If you are being offset onto your outside leg on a circle, it's centrifugal force. I think it's really very common for riders to experience what you are explaining, and end up putting more weight into the outside stirrup.
One thing that can help is to think of your inside thigh as really running down the inside of her body, and imagine it kind of holding you there. And since you should advance your inside hip a very small amount, if you think of your energy going down this inside thigh, agains the barrel and right behind the horse's leading shoulder, this will help you stay kind of , how can I say, "forked" over the horse's back evenly?
more riding with the whole seat (seatbones and thighs), less riding from the stirrups.