HOrses, at liberty , will usually not bend when going around a corner, in fact they will often almost counterbend (like going around the corner heeled over like a motorcycle). This is a very energy effeciant way of moving and requires the least effort on their part. It is also a very "forehand" heavy way of corner , with quite a bit of centrifugal force, which they counter by leaning inward to the circle/bend.
When the horse carries the rider, in order to not add a greater strain to the front legs and to build the kind of strength and balance to do manuavers at slower speeds, such as pirouettes, the horse must be taught to be more upright (not lean into the inside) and bring the combined weight of itself and its' rider back toward its' hind end. The focus is a more upward carrying energy, rather than a foreward "falling" energy , (as in running and leaping over jumps).