While bridges make tempting targets, destroying a bridge wouldn't help you all that much. Maybe you kill a senior officer or two, but you still have a largely functional starship that's going to be wanting to return the favor.
The TNG episode "The High Ground" features a more realistic scenario, but they weren't using transporters. Instead, it uses some sort of dangerous dimensional shift (the people who are using it are terrorists and very sick because of it). Shields are useless against the technology. At one point in the story, the terrorists shift aboard and plant a bomb on the warp core. It's a far more effective use of a transported bomb, because you can easily disable or destroy the ship there.
Another problem would be that you need to ensure the transporter beam is successful. We routinely see the transport process detected in the act. So if you picked it up, all you need to do is interfere with the process so you don't get a functional bomb at the end.
Finally, you're going to have a harder time beaming bombs as opposed to firing them. Even with ships from TOS, you could fire multiple torpedoes in the time it would take to beam them over.
As to bridge protection, the only time we see anything specific to protecting just the bridge is in Star Trek II, when they energize a defense field. Since we have no other canon uses, it's presumably just for TOS-era ships (in theory it should prevent transporter use). TNG-era ships had their bridges invaded by the Borg and the Dominion, but in those cases, their transporters could penetrate the shields. I think it's safe to assume they relied on the ship's general defenses as opposed to having something to just protect the bridge.
