Supress that!

Absolutely not. You should not win a sniper-engagement just because you shot in their direction. Nonsense mechanic that completely eliminates Snipers as a class of weapons that functions. The only people asking for it are people that literally want snipers to do nothing, ever. Why do you think you are OWED the opportunity to fight back when a sniper has a huge range advantage over you? Your weapon already beats them in ever other range. Why should you get a get-out-of-jail-free card for looking at them? If you are constantly getting pinned down by snipers, instead of coming to the forums, think about different choices you could make to address them/avoid them.  

Also, SUPPRESSION IS FINE!!! It prevents spawning and health regen. It actively makes a difference if you actually take advantage of it. We do not need inconsistent aim added to suppression, that just rewards the player that shoots first for no reason, on top of the advantages they already have from firing first. 

If you're struggling with a sniper, move. Stop standing in their direct sight-line with no competitive weapon to contest them. The maps do not tend to cater to snipers that much in this game, just find a different route or smoke your path. 

Why would only snipers need to flinch for almost getting hit? DICE keeps buffing automatic weapons with every new BF game and now they are far too accurate and deal too high damage, more then ever before, yet many players still keep complaining about sniper rifles, it's absurd.

And sniper rifles do have flinch when you hit them, it's not that much but BF never had much flinch for sniper rifles. And considering how OP automatic weapons are in this game, adding more flinch will tilt the balance even more in favor of non-sniper guns.

In fact, it would mainly force snipers to play like real snipers—hidden away, where even the slightest exposure would put them in danger—which is far from the case

Your comment doesn't make any sense. 
No one is talking about winning or losing a duel, since that would mean the death of one of the two players, and that's absolutely not what we're talking about here.
I’m talking about more suppressive fire that would allow for an escape. Suppression that would give the support role its full meaning—by firing at the snipers, preventing them from sticking their heads out, and thus allowing allies to advance without getting sniped. This would force snipers with sniper rifles to move around instead of just camping in the same spot for the entire match because they’re untouchable and the suppression is ineffective.

I hate to break it to you, but Battlefield games are not simulator type first person shooter games, it's always been an arcade type shooter. No single weapon in BF games is used in the exact same manner as their real life counterparts, and singling out only sniper rifles to be used like that is ridiculous.

Suppression IRL works so well because a single hit can mean death. In this game you can take a few hits and recover in a matter of seconds. Just because of this, suppression can never work the exact same way as it does in real life.

We absolutely had a working balance with suppression with the arcade style and it worked exactly as you would have hoped. Only when DICE started losing their most talented developers did we see the arcade style grow immensely. This is also about the same time that all the weapons started to become more accurate over longer distances and steep increases to muzzle velocity across the board.

As it is now, suppression generates a flinch on scoped weapons but only cosmetically as the flinch is more like a straight backward flinch that does nothing to impose an accuracy penalty. BF3 and 4 imposed a sway penalty. It was possible you still could land your shot but it was less probable.

I still think a lot of the gunplay changes were largely due to customer competition for other games in the genre. Not just COD but the entire genre altogether. 

Most players don't want some aim altering suppression mechanic where they feel they gotta hit the dirt. Add a sway penalty that builds up after successive rounds similar to the way recoil builds up. The first couple of rounds whizzing by may not matter. But cumulatively, they should impose a sway or accuracy penalty.

You should not win a sniper-engagement just because you shot in their direction. Nonsense mechanic that completely eliminates Snipers as a class of weapons that functions. The only people asking for it are people that literally want snipers to do nothing, ever. Why do you think you are OWED the opportunity to fight back when a sniper has a huge range advantage over you?

I would be a bit more nuanced than that.

First, a stronger suppression mechanic does not necessarily mean winning the engagement. Assuming you're doing it at a range your gun is not particularly efficient at, it merely forces the sniper to move away, unless they play exceptionally poorly and stay put.

Secondly, it really only applies to snipers you have spotted, and had time to align in your sights before they shot you dead. I really don't think this would result in snipers doing nothing, in fact, most of the time, when a sniper gets me, I did not see it (or saw them very late). 

I think at longer distances, the lack or presence of such a mechanic would make very little difference. Suppression works in a sphere of 5m around the target, you're not going to do that with any efficiency beyond 100m. It would be a way to get snipers to be more on the move when spotted, but it's not going to kill any of them. For me, the question is more about distances shorter than that, and some of the really infuriating situations that can come from engaging at those distances.

For me, the typical situation is the sniper popping up 30-40 meters away, not necessarily recognizing at first what weapon they're holding, putting 3-4 bullets in them, they "slowly" take their aim and then BAM, one shot to the head and I'm gone. The simple matter of the fact is that most people find this both infuriating and jarring. I guess the feeling is that at that distance, the sniper should be at a disadvantage, and it's not really the case now. 

The maps do not tend to cater to snipers that much in this game, just find a different route or smoke your path. 

I would strongly disagree with that. Sure, the smaller urban maps aren't great for snipers. But between that one that has a big mountain in the middle, that one with a big tower than overlooks half the objs, that one with a large rise on the side, the one with towers in the middle and rises on each HQs... Snipers are well catered for.

And a kind reminder that smoke is hard to get: not only is it only available only for support, but as that class is useless against tanks, they're much rarer in this buffed tanks world (a bit of a cascade of consequences here). 

A strong suppression mechanic does mean winning the engagement if you are at all of comparable skill level. People who played BF3 know this. Worst case scenario it legitimately brings the gunfight down to pure luck. You say "it should force them to move" but why? Why should one person 100 meters away be able to force a sniper to move with a rifle? That's insane given there are already limited long-range sight-lines where a sniper can actually impact an objective to begin with. Move where? To CQB where they're useless?  

Your examples of why snipers are unhealthy are quite poor. "The typical situation is a sniper popping up 30-40 meters away and killing me" First off, 30-40 meters away you have the advantage or are at least on even footing against snipers with most full-auto weapons. You should not be losing these engagements regularly. TTK at ~30 meters for full-auto weapons in this game is INSANE, and spread mechanics are very weak in this game. At 30 meters you have a really good chance of winning. 

If you're losing to snipers a ton, move more. Straight up. Move more. Strafe. Jump around. They can't handle that as easily and you can output more damage while moving erratically than they can. 

As for maps, I have no idea what maps you're talking about with "big tower that overlooks half the objs". Manhattan Bridge? That tower overlooks NO OBJECTIVES. Literally none. You cannot see any other capture point from B. If you're talking about Sobek City, we already know that map is getting reworked, not really relevant for this discussion because everyone knows it's bad. Same with Blackwell Fields. Liberation Peak, sure that's decent for snipers. If you're talking about Firestorm, A, C, and B have near total cover from Snipers most of the time due to the structures on the obj, E is extremely far out from most sniper locations and allows you to LoS snipers on C, and D also has a whole building to hide in. You can get people running around like chickens with their head cut off out in the open, but you will not be a meaningful or oppressive force on this map against anyone with a brain if you're camping up in the tower. I can't remember the last time a sniper up on the tower at C killed me, and Firestorm is like the most common map I get in rotation. And there's 3 helis per team....

Forgot about Mirak Valley, maybe you meant that one? In that case B and D are completely enclosed, C is super far from the central towers and is completely LoS blocked by the house, and A and E are visible from the towers but also have some individual cover that can protect you if you play around it. Snipers can get stray infantry running across the fields, too. That all seems very healthy. 3/5 objectives are not visible from the towers, the other two that might be have plenty of cover, there are tons of helis and jets on the map, and zip lines that infantry at B/D can take to go directly to your location and take you out in CQB. 

I was merely trying to outline to you why people may want a better suppression mechanic, and nuance your quite entirely one-sided response. You may note I did not actually take position for or against it.

Personally, I DO think snipers are in an unhealthy position, but for me it's entirely linked to the general ease of sniping compared to previous titles. The rangefinder is a big part of this, as are the flatter ballistics. I've outlined elsewhere why I think that is unhealthy, I won't repeat it here. But I don't think increased suppression would help in this particular regard, and anyways I wouldn't link that mechanic specifically to sniping: for me, suppression would be mostly important to give LMGs the meaning they are currently lacking.

Edit: I also wanted to add, I see too many of these discussions going as if somehow there were snipers on one side, and non-snipers on the other. Sorry, but that is mostly not the case, most of us move from class to class and gun to gun: I don't think sniping is a good place, and I say this from a position of not just having been on the receiving end, but also having actually played it quite a bit.