Easiest way to beat him is to just stand right next to him. You'll see three skulls appear in a row low to the ground. Do an upwards dodge to avoid his axe swing. He'll do the same attack both times (as long as you stay close to him). Dodge the next two in the same way.

After the third attack, he'll start getting ready to ram you. Once he goes into that animation, get up on one of the platforms. He'll still do the attack and slam into the wall, stunning himself. Once he's stunned, get down there and dish out some damage! The best attack to do is to hold down punch while in slow motion and zoomed in. This is the strongest attack in the game, and does great against stunned bosses.

Make sure to stop hitting him once he comes out of his daze and starts blocking your attacks. If you hit him too much when he's not dazed, then he may do a HUGE attack (signified by a large axe) that you can't dodge, and takes off a ton of health if it hits. If you see the large axe, get out of there. Also, if you hit him too much while he blocks, then his axe will break, causing him do do more jumping attacks with missiles from his desk and bombs from the ceiling, which is just needlessly difficult.

If your health gets low, use mach-speed attacks on him while he's stunned to break the boxes in the background for burgers.

Like almost every other enemy and boss in the game, it's another case of dodging until you get a good opening. Don't even try to land any attacks in the beginning of the fight. This will just cause him to counterattack and won't do a whole lot of damage. The trick to easily beating Bruce comes from using his swimming bite attack to your advantage. I've seen that a lot of people have trouble avoiding it, but it's really easy. All he'll ever do is charge straight from his current location to your current location. So if you know he's to the side of you, get ready to jump, if he's above or below, get ready to move to the side. After you avoid him for a while, he'll get tired and start to wheeze. Run over to him and do the good old slow-zoom-punch move. After you hit him a few times he'll dive into the pool of water and start wheezing some more. Make sure to follow him. If you don't interrupt him down there, he'll regenerate health. Hit him some more in the water to stop his regeneration and he'll most likely raise the water level.

The strategy for the raised water level is the same, except his attacks differ. Keep your distance and dodge until he spins and mines appear everywhere. Once the mines appear, he'll always do his biting attack next. Position yourself so that a mine is directly between you two, and he'll run right into it, getting it stuck in his mouth. Go into slow (and zoom, if you've got the juice), and hit the mine. The explosion will do a good amount of damage, and you may be able to get a few extra hits in.

Just keep doing this and you'll have him beat pretty easily. It's also good to learn how to use the Ukemi move to recover some health if he does get you in a bite.