It's a LOT easier if you...
1. Keep an eye on the bird and AS SOON it goes into it's animation where it spawns minions, RUN LIKE HELL to the other side of the battle field immediately. This is because they always spawn near you. Relocating as soon as it starts going into that animation will give you time to distance yourself before they appear.
2. Once distanced from it's minions, they are scripted to sling arrows vs rush you. Keep an eye on the bird while running back and forth at that end of the battle area, while avoiding it's fire balls and it's minions arrows.
3. The 3rd attack is the bird's close range attack. This is when you want to just let it come to you and ready your bow to shoot. As soon as it stops to start the attack, aim the bow and when it starts slowing time it will be easy to shoot it in the mouth from close range.
4 After the 3rd set of same rinse and repeat procedure, the bird will drop and you need to send the tiger to it. After the tiger jumps in it's mouth it will again open it's mouth and you can get in another shot. Not sure if this really helps but why not.
Just do the above 3 times and the bird is toast. With this method it's conceivable you may not even have to heal, esp on Easy. I didn't even on Normal. I took a few glancing arrows, but only enough to drop one of my health bars halfway or so. The tiger also stays with you and healthy this way. It can be done in well under 5 minutes. Basically the time it takes for the bird to go through it's 3 phases times 3.