i would get the wheels balanced. but if its that bad, it sounds like a bent wheel to me...

 

Wouldn't a bent/unbalanced wheel/tire cause vibration or torsion on the steering wheel and not bouncing? This sounds like the steering rack might be loose. Put the car up on a rack/jacks with the wheels unsprung and try to bounce the steering wheel up and down. Also get under the car and grab the shaft coming from the steering rack, and push and pull on it. See if you get any play.
The bushing in the steering column could be shot as well. I believe Whiteline makes a poly one.

 

I think it's Jesus. He's saying, "Slow down."

 

A summer or two ago I had the dash apart including the steering column out. I remember the stern warning in the service manual about not releasing the tilt mechanism with the column out of the car or the column is toast. I made sure I didn't do this but I do not know what happens if you do. Have you had the steering column out of your car?