On screen during the series, we see them catch a total of six bounty heads in four different episodes.
The first time they successfully catch a bounty was at the beginning of episode 8 with the hilariously named trio Huey, Dewey, and Louie. At the end of the same episode they catch all but one of Piccaro Calvino's crew (since Roco died in the shootout and even if he hadn't, there was no way Spike was going to turn him in).
Next was episode 10 with another two bounties. First was Baker Poncherero who's already tied up on the Bebop at the beginning of the episode, and then at the end of the episode, Jet turns in Rhint Celonias, his ex's new boyfriend.
The next bounty they get isn't until episode 15 when they turn in Faye's old flame Whitney Hagas Matsumoto, though they're disappointed by how small the reward was.
The final bounty we see them catch is in episode 22 when Faye and Jet catch Teddy Bomber while he's fleeing away from Spike and Andy's fight.
Although those are the only ones we actually see them catch through the whole series, and let's be honest, a show about bounty hunters who only actually catch bounties in 4 out of 26 episodes (AND don't even catch any in the movie either) doesn't SEEM that impressive, it's pretty heavily implied that they catch a lot more offscreen. In particular, Teddy Bomber singles out Spike Spiegel as one of the most feared bounty hunters in the whole solar system, so I would have to think that they're a lot more successful in the times where we don't see them. But the show honestly is much more humorous, entertaining, and realistic in showing that anything can happen and failure is something that happens often when tracking low life scum across dozens of planets, asteroids and moons, and when they fail, all there is to do is move on to the next bounty.