When you consider that the bridge is on the top and center of the saucer section and a very enticing target the constant explosions make a bit of sense. It's not that the enterprise got shot in the ass and it sent a current up to the computer that monitors shift rotations, no the high powered energy weaponry is aimed right on the other side of the wall. Structural integrity fields, shields, ablative armor, and other technobabble keep the the whole bridge from popping like a balloon when something bad happens.
It's like if your tv is plugged into a house with breakers and safely on a surge protector it's not going to keep your tv from getting fried if zeus decides to target that outlet specifically and strike it.
THEY DO HAVE THEM! Every once in a while a trek will show seatbelts and then proceed to forget about them later.