I've dealt with a lot of horses that have done this as well. My mare always anticipates turning the first barrel and I have to do a lot of work to keep her off of it.
The main thing is, don't forget your basics! Don't always run your horse. You horse will become hot, hard to handle and it won't make barrel racing fun to deal with a horse like that. Plus it takes a lot of retraining to break their bad habits once they start.
With my mare, I when I practice, I run her maybe once or twice at a fast lope through the pattern at home. That's it. The rest of the time I walk, trot or slow lope her. Whenever she starts to dive into the barrel, I drop her down to a walk and don't let her run. I don't get worked up about it. She's a sensitive mare and feeds off my energy so if I get upset with her, she gets hyped up and won't listen.
I make sure my barrel horses know how to walk, trot, canter, rollback, stop instantly and sidepass too. Sidepassing is great for barrel and pole horses! It helps with their lead changes and creates a super balanced horse.
Anyways, when my mare starts to get heavy on the barrel, I simply go past it instead of turning. She wants to turn, so I don't let her. Say I'm turning right around the first barrel, she starts diving, I pick up my outside rein, put my inside leg on her and just go past it then do exercises to the left. Get her thinking about something other then that barrel. Then I'll set myself up for the first barrel again, and simply ride by it again. Eventually once she's balanced again, I'll let her turn around the barrel. That's her reward. That's what she wanted. :)