I have found a fix for this! I am also using an xbox 360 controller.
First of all, some of the input delay I was experiencing was actually reduced by lowering some of the graphics settings. (Which I found surprising. Most games, even if I raise graphics settings to the point of in-game stuttering, the controls would still be spot on, but not so on this game. Reducing shadows and other settings to lower levels actually helped a lot.)
But I still found that changing direction of a turn was painfully, unrealistically slow.
I fixed this by going to www.tocaedit.com and downloading: NFSMW2012 Realistic Plugin.
As well as x360ce, which is a 360 controller emulator. (I know it's a weird thing to download when you are actually using a 360 controller already.)
Follow the instructions for the realistic plugin (checking the box for raw input is what actually does it) and in game now it turns really well... Too well actually, as I can now go left to right to left instantly. Whereas in a car it would take a moment to turn the wheel. But it's waaaay better than before which was like a granny slowly turning a huge boat wheel.
The realisitc plugin does other things but I haven't checked it out yet. Also I am trying to tweak it to get things more realistic but not sure if I'll be able to. Either way it is now much easier to control the cars. But watch out as you can now go left, right, left, right and spin out. :D