A horse only gets full speed by going to a gallop. This is when all four legs leave the ground at the same time. Engineering a device to turn gallop into moving wheels would be very difficult. About the only thing that would work would be the conveyor belt gizmo shown above. It would have to be large to support the weight of a horse and to take the pounding of a full gallop. Plus it would have to have 4 wheels or some sort of device that would balance a two wheeled vehicle for the horse. Probably too heavy to generate any real speed.

My guess if you matched a horse powered vehicle (where the horse was a passenger) against a human cyclist, the human would win every time.