had this on an altered, it was in my view a culmination of play in the whole steering assembley, exaggerated in my case by fitting taller rear tires and losing some castor. Kingpin bushes had some play, rods ends were old, and steering box had some play too. It all gets magnified back at the wheel, i used to get it coming out of a rolling burnout (off the pedal)
There is a technical description as to why it happens and it's related to the gyroscopic effect a spnning wheel has but I've not delved into that too much. A damper does mask the problem, but make sure all steering components are as they should be, and the wheels / tires are balanced up and pressures evenly matched.