TWH's can walk as fast or slow as you want. It's just a walk, like any other breed. It's the trot, or for the TWH, the running walk that is super fast. They don't trot slow. They will trot as fast as most other horses canter. TWH's can also canter... But on just the plain flat walk, they just walk.
I used to trail ride a TWH and my Dad was on a mustang mare. No problem.
The only difference was that particular mustang had a rough gate regardless of the gate, and my TWH was smooth at whatever gate. Hehe. I used to tease my Dad about this.