My 2003 buick century right turn signal flashes fast but only in the front. I was told that this is the sign that a bulb is burnt out, which it was. The right rear bulb was burnt out so I replaced it and I still have the same problem. The hazard flasher does not work on that socket or the turn signal. But the brake works and the running light works in that socket. What could possibly be the problem. Thanks for your help.

... Possibaly the relay could be going bad... have you checked it... not sure where it is though, and I know eariler models had one for left and one for right
Relays will sometimes will act funny when they are going bad.