I've been having this problem for the past month or so without actually figure out what causes it. The problem I'm having is that at random times, if I scroll through and use the back-up menu and maybe other menus as well (eg. Police Radio), the car steering with the controller breaks and I can't make precise steering moves anymore. It still turns left or right if I move the stick all the way in that direction but if I move the stick only a fraction of the way, the wheels will actually move in the opposite direction for a split second steering me directly into the object I try to avoid. 

Since it mostly appears after requesting back-up multiple times, it means the steering goes haywire during car chases which is quite frustrating. The only solution I've found that fixes the problem is to stop the car, exit and enter the car again. But I could never pinpoint the exact cause of this bug and I can't replicate it on the spot.

Has anyone else encountered this problem? I'm using an XboX One controller.

I have this problem all the time. I've never figured out what causes it, usually to fix it I have to exit GTA 5 and come back. I use a wired Xbox 360 controller.

That's unfortunate. So it doesn't fix itself if you exit the car and enter it again?