King Bulblin must be hit 8 times in Phase 1 and 2 times in Phase 2.
Pull sword and dash to approach King Bulblin's right flank while charging a spin attack. Release the spin attack when near enough, then continue slashing. After he has been hit 3 times, King Bulblin will either turn slightly right or sharply left. If he turns right, continue to slash him without dashing; if he turns left, dash quickly to follow him, charge and release another spin attack, and then continue slashing. After 3 more hits (6 total so far), he will turn again. Repeat the previous steps and hit him 2 more times to finish the phase.
Tutorial on the basics of a good fight: here
With the Clawshot, it is possible to shoot King Bulblin from a distance to hit him without triggering him to start moving. Start by immediately turning Epona around and quickspinning all the bulblins when they get close. Backflip off of Epona, walk about 3 steps back, then aim and shoot King Bulblin 8 times to finish the phase.
With the Bow, it is possible to shoot King Bulblin from a distance to hit him without triggering him to start moving. As the fight begins, target him and shoot two arrows (continue to target him through the entire phase). Hesitate briefly between shots to account for his invincibility frames after each hit.
After shooting the 2nd arrow, continue holding L and press and hold B to charge a spin attack. Release the spin attack when the first two boars get close enough to Link, then perform a quickspin to kill the next wave of bulblin riders, being careful not to move Epona forward and cause King Bulblin to start moving. Shoot King Bulblin with 6 more arrows, spacing them out as described above, to finish the phase.
Dash 3 times, then hold down to turn Epona around once she passes the large crack in the bridge. After turning, hold B to charge a spin attack, and move to the right side of the bridge when King Bulblin gets near to avoid being hit. Release the spin attack as he pulls up beside Epona, then hold left and slash him once more to finish the fight.
Dash 3 times, then hold down to turn Epona around immediately. After turning, enter first person, pan the camera 180 degrees, and shoot King Bulblin with the clawshot. After the cutscene of him being hit, shoot the clawshot again immediately to end the fight. This saves 1 second versus the fight with sword, but it is much riskier and more difficult.
Shoot King Bulblin twice with arrows. He begins far enough away that aiming to shoot as soon as possible is difficult, but dashing once, immediately nocking an arrow, and shooting as soon as the shot is fully charged will consistently hit King Bulblin. The second shot can also be straight ahead.
Dash with Epona from the starting position, but let her dash run out before reaching the corner where the clip will be performed. At non-dashing speeds, Epona can be steered directly into the corner without her sliding along the wall to the right. Align Epona roughly parallel to the ledge on the left, then walk her forward until her head is just past the corner. Backflip off of Epona and roll/run under her neck to the corner. Epona's head will turn toward's Link, and if she is in the right position, she will push him out of bounds. It can help to have Link moving slightly (holding ESS into or near the corner) or climbing the ledge at the far right side. Voiding will cause Link to respawn on the other side of the bridge as though he had just defeated King Bulblin.
It is possible to skip the entire fight with King Bulblin 2 on the bridge if by voiding out anywhere when the minimap is active near the bridge however this was long thought to be impossible due to there being huge midna triggers and not being able to bomb boost over fences. However it is possible to get OoB at the very start of the escort and then with a bomb-boost from a backslice getting further OoB and be able to roll down to where it is required to void out. The method requires you to get onto the wagon which is rather close, once you do this as soon as the escort starts you have a small window to LJA over onto the cliff where it isn't slippy, another unfortunate side-effect of using the wagon is you only get one attempt at the start of the escort. The best setup so far is outlined below. This trick saves around 50s in 100% currently.
Hold L+up out of the cutscene and go through the text. Open up the item menu as soon as Epona starts to dash, hold down and close the item menu again. Wait for Epona to stop and then hold down left to turn.
As soon as Epona stops turning, press R+A to backflip onto the wagon. Now you want to LJA towards the cliff and land where you can stand, this is a really small window one recommendation is to turn the camera while link is getting back up, also use C-up into boomerang for the extra speed. The targeting needs to be rather quick, also the jump is not very precise.
Now once you are on the cliff stand as high as you can and try to find a spot that minimizes the distance between you and the top. From here turn away and sword drop a bomb so you can then throw the boomerang and catch the bomb in it, now time it so you start your backslice up the cliff as the bomb is coming back, if timed correctly the bomb will damage boost you at the peak of the backslice and you will get on top of the cliff over the seam. From here just roll towards lanayru bridge until you jump through the floor and void out in the right area. You will now respawn after the fight.
If Bow is not available, one can throw bombs at specific times to defeat King Bulblin 2. It takes 3 bombs to defeat him if King Bulblin 1 was not defeated, but only 2 bombs otherwise. The bomb throw timing is precise because his hitbox is small, though it also extends sideways somewhat.
This video shows when to pull the bombs without buffering:
This video shows a way to buffer the bomb throws in case that helps time them:
Hit the trigger for the King Bulblin 2 fight as wolf. After the cutscene, turn around, dash off the bridge to the spot shown in the video below, and wait. When King Bulblin appears, dash to the left side of the wagon and transform into human. Position Link next to Telma's horse on a stone slab in the ground and face the corner of the wall. Once King Bulblin runs into the wall, use the Boomerang to hit him with 3 bombs, aiming the third one significantly higher than the other two. Climb Epona after the third hit to trigger the cutscene.
This is 3 seconds slower than the unbuffered bomb throwing strat above.
Jump attack + quickspin King Bulblin to make him slump down. Quickspin three more times to finish the fight.
The trigger that loads the path to Arbiter's Grounds doesn't exist until Link has defeated King Bulblin and broken through the fences with a boar.
Trigger the barrier, then continue to trigger King Bulblin falling. Skip the cutscene as soon as possible so that King Bulblin will become vulnerable as soon as possible after landing. When he begins his attack animation, do a jump attack + quickspin combo followed by three additional quickspins. Backflip to dodge his swing, then jump attack + quickspin once more to finish the fight.
With Greatspin, quickspin 4 times after the first jump attack + quickspin to finish the fight without having to dodge an attack.
When King Bulblin begins his attack animation, do the following:
The quickspins' directions are important.