You should listen to me. No, seriously, listen to me.
Join Date: Oct 2008
Location: Santa Barbara
Motorcycle: Blue 2006 GSX-R1000
Re: Electrical issues with my GSXR
Download a service manual. There's a wiring diagram at the end. I've downloaded the K6 600 manual from somewhere and it should have the same diagram as yours or very close. The position, tail, license plate, and turn signal lights all get power from a Br lead that comes from the ignition switch. Try turning the switch to P and check if the lights come on (Br then gets power nearly straight from the battery). If so, the Br wiring is OK. In ON, the power to Br comes from an O/G lead and the IGNITION fuse. The horn should work whenever O/G has power.
"My tail light only works when the brakes are applied"
I suspect that what you're seeing is the brake light.
Alternately you have some sort of B/W ground problem. But since you're running, I'm less suspicious of it. Fix the lights and then see what the backlight is doing.