(I doubt you're still stuck, but just in case someone else has trouble...)
At the end of the race, it helps to get a little aggressive with the other Gorons. When they hit you, it slows you down, but if you hit them, it will slow them down. Earlier in the race there are too many obstacles and such to worry about doing this, but when you get close to the end, it will seriously help if you knock the others around a little.