The turn signals and the emergency flashers each have their own flasher. Look on the fuse panel up over your left foot. You'll see two "thingies" about 1-1/4" in diameter and maybe an inch to 1-1/2" deep sticking out of the panel. One is at lower left, the other is at upper right. I can never remember which is which, but pull one out -- if the turns don't work, that's the one.
First thing to check is if you have a burned out bulb on the right side. The standard flasher (for non-tow equipped XJs) is load-sensitive. If a bulb burns out, it either doesn't flash or flashes very fast as an alert.
If no bulbs are burned out, just to test find the one for turns then swap in the one for the emergency flashers. If that solves the problem, just buy a new flasher unit. If that doesn't solve the problem, look for a possible poor ground on one of the lights on the right side.