It certainly isn't tire/wheel balance -- this would not be the cause on a curve without also being noticeable on the higher speed straights.

Given that it shows "when I hit a certain far degree" and also when reversing at full lock (i.e., "completely cranked"), I would suggest you are in bad need of a wheel alignment shop's services, not so much to check your straight-ahead specs but rather to check your front wheels' alignment when the wheels are turned.

Front wheels aren't supposed to remain parallel when they're turned. The suspension is designed so that, when the wheels are turned, the inner and outer wheels are supposed to follow slightly different arcs (or paths), the inner one slightly tighter than the outer one. I'll bet that they aren't following their correct paths (i.e., aren't pointed in their proper direction), and the feeling you get is due to the two front wheels fighting each other, each trying to push the other sideways.

Does the effect abate when it's raining (wet roads)? That would be because the two front tires have less traction, and their slight sideways slippage mitigates the struggle between them.
Also, have you noticed any unusual or non-uniform tire wear?

Possible causes include steering and and tie rods misadjusted or bent.

Also, possibly your ball joints are failing -- they're subjected to heavier forces when you're turning.

And finally, wheel bearings (but you haven't mentioned any noise). Hmmm.

Good luck.