It's nice to see some good pics of FZR's. Here are mine I'm fixing up. The yellow one is a 96 600 with D&D pipe, polished wheels, stage 3 jet kit, airtech body work. The black one is a 91 1000 with 94 body work, polished frame and swingarm, yosh exhaust, 1040 kit.
Sounds like it could be the head bearings. Put the front wheel in the air and pull on the forks to see if there is any play front to back. It may just need to be retorqued or you could replace the bearings with allballs tapered bearings.
I've seen them too. Gotta get a battery for the bike to enjoy this nice weather. I have nothing to say to them cause they won't listen. I just shake my head and stay away from them.