Which jousting team was it out of interest ??
Quite a few jousting teams use polo ponies for the displays, as they are quick off the mark, have good brakes and turning etc. Obviously using the same tack they would do normally.
When jousting you don't need any extra reins to worry about as well as the lance, shield etc. Also you should have noticed that the majority of the horses would not have been ridden on a tight rein as they know their job ie keeping close to the tilt, galloping toward another horse on the other side of it, and that the riders will only apply rein pressure when requiring brakes, or turning at the end of the tilt or for any stunts that might be done etc
Debs