Agree with this... If the issue gets worse by going faster, it's an imbalance somewhere. The turning is largely due to transferring the "force" onto other parts of the vehicle, so it's still happening, but you won't notice it as much.This sounds like a drive train issue. I would look toward the front CV axles especially since you mention it changes with steering input.
Anything that rotates should be checked, tires, rotors, CV axles, bearings. (if you feel it mostly in steering wheel vs the back of your seat, it's more than likely front).