Any ideas on preventing or keeping the reins from flipping over the horses head during the run? 

I don't know what your reins are  made of that you are using. Using leather reins might help. They are heavier and would give you a better grip on them. This might keep them from flipping over. 

Take your snaps off and attach reins directly to bit. The swinging snap is what causes the flip. Thanks for that nugget of knowledge Uncle Ed. Haven't flipped a rein since.

Thank you soooo much!  I use heavier leather laces reins with snaps. I will remove the snaps per "uncle Ed's" advice. God bless him. He still continues to help us out!!

File size heavy rubber band around reins, a few inches behind the snaps or a piece of string tied the same way

Shortening your reins usually helps. 

But if not look at what Tiany Schuster did with Shomance. 

Cable reins seem to help with this issue. They don't flop around like traditional barrel reins.