Well thank goodness I'm not the only one who has had massive problems with that part. I was more frustrated with the fact that the carriage kept turning around when the bird was dropping those bombs. I finally figured out that is where the bombs were coming from, so I ran ahead of the carriage, let it burn until I was able to shoot the bird out of the sky with bow and arrow + Z-Targeting. Then I put the fire out on the carriage.
The first time I made it all the way to the end with 1/2 heart left and right as I was going to open the gate, I got killed and had to do it all over again.