is this for direct reining riding? like english or dressage?
my understanding is that the inside rein will ask for the flexion to the inside of the turn and get th horse thinking in that direction. the outside rein, laid against the neck and fairly steady, support the horse , and backs up the suggestion to bend to the left, which is also created by the legs.
The outside rein sets the limit of the bend, which is initiated by the inside. So the inside rein says, "bend this way , or rather flex this way and think this way" and when the horse starts to flex his head that direction, the outside rein will allow the horse to flex only as much as is appropriate for the tightness of the turn. But, the outisde rein will encourage ther horse to move his shoulders over that direction. The reason he doesn't just fall sideways that way is that your inside leg is moderately firm , asking him to not fall inward, but to bend around the leg, and go forward.
Try riding some squares. go to the corner and when you make your turn, visualize almost lifting up your horse's shoulders and swinging them onto the new straight line. The lifting happens with the outside rein. I don't mean really lifting, but this is just a visualizing excercize to get you thinking on turning from the outside.