there are all sorts of bits you can use, you ort you try him/ her in some before you go spending money as they are expensive you can get mylers cherry rollers etc, which play with different parts of the mouth which may work or a 3 ring gang, a waterford or a pelham i will advise you to flat in a snaffle and jump in the stronger bit

I don't know if your still having this problem but thought i share with you my experience from my strong welsh cob. When we got him he was in a gag on the second bit. Gags usually put pressure on the poll and it made him put his head down when i was trying to stop him taking off with me, which he took advantage of and once his head was down he was off!!. My instructor told me to switch to a pelham which worked as it didn't make him put his head down. He once again had a nice head carriage that was comfortable for him and didn't take off as much. Then though he got clever and decided to lock his jaw and take off with me that was. I checked with the dentist and his mouth was fine so a friend of mine said to put a flash on. Since having a flash and pelham he is a lot more comfortable and relaxed and i don't have to fight to stop him taking advantage and zooming off!  Hope this is helpful to you or anyone with a strong cob like my Prince :)

I'd start with the basics again to be honest.

But I don't know your mare in person. I use a rubber three ring gag on my Highland, Arab and with it having the rings, I change it depending on what mood she's in :)