Configure the analog stick on the PS3 controller as “vehicle throttle” in GTA3 on Mac

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.

The N64 analog stick is extremely unusual. It does not use potentiometers, as most other platforms (i.e. playstation controller), but instead used a geared-up digital incremental rotary encoders.

^{This is also why you have to have the control stick centered when the system turns on. Since the stick mechanism has no concept of "centered", the N64 assumes that whatever position the control stick is in when the system is turned on is "center".}

As the gearing wore down, the play of the controller progressively increased.

Wikipedia:

The N64 Analog stick does not use analog potentiometers. It uses light emitting diodes and photo detectors controlled by sensor wheels. The sensor wheels are plastic hubs which produce a shutter effect allowing for an accurate direct digital read. The sensor wheels give direct correlation to the stick position compared to potentiometers which can change resistance values over time. However gameplay function of the stick itself is not on or off digital, but analog in a sense, the more the stick is pushed the faster a game character would walk to run.

The analog stick was prone to some long-term reliability issues. If used excessively, the stick became "loose", which means it will not fully return to center position, which may render gameplay more difficult by giving unintended, non-user input to the system. This loosening can be caused by rotating it intensively - a common practice in games like Mario Party where it offers an advantage for some mini-games.