I wouldn't start seriously running barrels until she was at least five. It's too stressful on them and leads to breakdowns. I start all horses out on barrels at a walk. But before you should even break out the barrels she should respond to leg pressure, move back, sideways, and forward, STOP, and lengthen and shorten her stride for you.
I don't like tom thumbs. I think at this age they should all be broke in a snaffle. I use a greg darnell or a gag bit on my barrel horse but he is finished and I can run in a snaffle if I want to and he will stop.
Don't up the harshness of the bit when your horse won't pay attention, it just leads to more problems.
When she starts to want to run away from you, do circles. Start off big and make them smaller as you go. Just give her something to put her mind on. I know barrels are supposed to be running really fast and exciting, but if you want your horse to last you need your basics to be very very sound.