Download Origin on External Hard Drive

@makerunurself  Just so you know, the game will likely run more slowly on an external drive than it would on an internal drive.  This depends in part on the connection speed of your external, and in part on what type of drive it is, but you will be sacrificing some performance.  If you're okay with that, it's certainly doable.

There are two separate sets of data: the game's program files (the game itself that gets installed from Origin), and your user data (your saves, mods, custom content, etc.).  Sims 4 will always look for its user data folder in Documents\Electronic Arts, and if no folder is there, it will create another one.

Installing the program files on your external drive isn't complicated, and maybe you've already done this.  It's best to keep Origin itself on the internal drive though, and not usually a problem since it takes up so little space.  If you have actually moved any Origin files, please uninstall Origin completely and reinstall it from scratch on your internal drive.  That means deleting anything you've moved to the external drive, then uninstalling Origin "the hard way" and reinstalling.

https://help.ea.com/en-us/help/origin/origin/manually-uninstalling-origin/

If you never actually moved Origin or any of its files (not the game, just Origin), you can skip this step.  But if you're not sure whether all its files are intact, it's better to uninstall and reinstall from scratch.

For the Sims 4 program files, you can change the location when you install, or you can (usually) tell Origin to move the files itself.  Be sure to make a folder on the external drive into which the game will go (don't name it Sims 4 or Electronic Arts), as it's usually not possible, and definitely not a good idea, to install the game directly onto the drive without using a folder.  If the game is still on your internal drive and you want to move it without reinstalling, open your Origin game library, right-click on the Sims 4 icon, and select Move Game, then pick the folder you've made on the external drive.  If you need help with this step, just let me know whether the game is still installed and whether you've tried to move any files around.

For the user data folder, you can create a symbolic link to redirect from your internal Documents folder to the external hard drive.  It's fairly straightforward in Windows, although you need to be confident you can follow the instructions to the letter.  Since you're only moving your user data, not the entirety of Documents, there's not much risk other than the possibility of losing your saves.  Be sure to create a backup of the folder elsewhere before you start, and you'll be fine.

https://crinrict.com/blog/2020/02/moving-windows-documents-folder-to-external-drive-via-symbolic-link.html

Again, if any of this is confusing, feel free to ask for help.  But please describe the step you're trying to complete, and where the files are currently located.