Your description of the speed and effect sounds like "wheel tramp", which I've only experienced once, as a passenger in a kingpin-type front end 1953 Ford sedan in 1960. It would spontaneously begin at 35-40 mph on a smoothly paved road and was amazingly violent, making the steering wheel rotate back and forth and the whole front of the car jump around. A sudden swerve by the driver to the steering wheel could calm it briefly. My friend who was driving had been told the car needed kingpin bushing replacement.

I read somewhere decades ago that there is rapid dynamic wobbling of the front wheels in the left-right steering direction that at the 35-40 mph speed sets up gyroscopic precession forces that make the tire bounce up and down on the road and the effect reinforces the steering wobble in a "vicious circle" of instability, like a flag waving in the wind.

Car speed for tramp is lots slower than the 65-70 mph that purely vertical tire bounce from out-of-round or out-of-balance typically occurs at for independently suspended front wheels. Looseness in suspension and steering parts including tie rod ends is no help in avoiding tramp's onset.

Search "wheel tramp" online. This SAE paper abstract has a brief summary: http://papers.sae.org/330042/

If your rear tires are not unevenly worn, you might try them on the front to see if tire tread wear profile makes any difference.