My pony is taking turns very wide, and Im having trouble fixing it. Help?

My barrel pony used to have a lot of trouble giving to the bit. I took her off barrels because she had absolutely no give or softness, and was frustrating to ride because of that. Ive spent the last 2 months on a snaffle teaching her to flex, collect, and give. But now then I turn her around a barrel, she overbends towards her girth, and her hindquarters swings out, making a wide and unbalanced turn. How do I collect her hind underneath her while still maintaining speed? Im not going to rush out and gallop her every time I get on, but Id like to be able to take her up a notch one day, but still have a clean pattern.. Details so I can get a detailed answer... She gives well to leg cues, rides on snaffle, short flexible neck, 13.3 hh, very high natural head set, I have to use a tie down(I will not take out the tie down.. I know some people have a problem with it, but she has broken my nose before with her head). Hope I gave enough info, I really do need help on this one.