02 gsxr 1000 headlights turn off n on while riding while hi beam works wen I turn it on

It could be driving lights mode instead of low beam. Check connections.

check the plug earth wire for clean fastening to earth. On a H4 bulb one wire is power for power for low beam the other is power for high beam and the third wire(normally black) is the earth wire.. Place a continuity tester ( multimeter set to read resistance ) on the black and place the other lead on the bike frame and see if you get a circuit. If not check that wire all the way. It is possible for the current to flow from one beam back through the other beam hence stopping current flow because the resistance is excessive in the earth wire.

Yes you should have a headlight realay. You can find where it is, get new on at rommiessusuki.com. You can also get wiring schematics also.

Steering damper

Low beam switch may be bad. use a meter to check for power to lowbeam side of lamp. cleaning with electrical contact spray may help.

Sounds like your low beam relay isn't working, try swapping it with the high beam relay to see if it cures, if it does pop down to a Suzuki dealer to get yourself a new one, if it doesn't then check all the connections in between switchgear and relay, relay and headlight for corrosion.

you need to ground your starter to see if is good or if it is your solenoild check all fuses first then take a rubber handled screw driver and touch the hot to the frame or a groun if the starter spins then your solenoid is bad or a fuse but more than likley the solenoid is bad

hi, i had a k2 gsxr from new, mine has always only run on one low beam light and two main beams.
so unless mine was defective when i got it i presume thats normal for them