Why does the rider fall in the forward direction when a running horse stops suddenly?
The lower part of the body (including waist and legs) is in contact with the horse while the upper part is free. This is why only the upper part of the body is project backwards or forwards. The center point of the body acts as a pivot around which the upper body turns.
The tendency of the upper part of the body is to remain in motion due to which it continues to move and thus the rider falls forward.