Change the steam controller’s sticks

One of the things I liked most about some of the Gran Turismo games is that they let you use both Playstation analog sticks (the left to steer, and the right for forward and reverse acceleration). I'm not sure if the games you're playing have an option for that, but I'd try it and practice that way for a bit. I find that it not only centers your hands better on the controller (i.e. your right hand isn't moving the left stick when you hit the accelerate button a bit too hard), but that acceleration and braking are much, much smoother.

It might not be the same problem, but most of my turning problems were solved by smoother acceleration and braking into the turns, not by the actual steering itself. Give it a shot if you have the option.

Other than that suggestion, try to practice not tapping the sticks so much, but keeping pressure on them at all times. When you start to initiate a turn, give yourself enough time to decelerate and gradually and smoothly steer toward the straightest exit through the curve rather than hammering the stick at the last second. Knowing the course really helps here, since your hands will start to remember what they're supposed to do at any given point.

I suppose that's pretty general, but I hope there's something useful in there for ya.

You can use x360ce to emulate an Xbox 360 controller with any old DirectInput controller.