I want in on the pool. I say its a tie rod or a steering gear. What you are describing sounds like what we 2nd gen dodge cummins owners lovingly refer to as "death wobble".
2002 Bandit 1200/ GSXR cams/ 1277 BB Kit/ Holeshot header and can/ 38mm flatslides/ a good head/lotsa hp/lotsa tq- lots of rear tires...