It's interesting that elevator bits tend to help with lifting the head... I actually have the opposite problem with my mare. She is VERY strong when jumping or out galloping (but not naughty by any means). With the elevator though, first thing she does is tuck her nose and lean into the bridle, taking away most of my braking and turning control. I wonder why she would have this reaction to the 3 ring bit?
Another person posted that she had the exact opposite experience with leverage bits. The reality is most due have this painful lesson with leverage bits because whenever you up the ante with a BIT you must up your ante with a lot more leg and more tactful aids. Many horses who get low in the bridle are telling the rider something, more bit only creates the false illusion of control and when horses curl and run off at shows with these 3 ring bits or heck look at poor Cindy Burge who died in a leverage bit when her horse curdled in Montana and fell with her in a wheat field unable to stop. My point is leverage bits are NOT the answer unless you can refine your skills!