I used to school two young 18hh Clydesdales, who at 4/5 yo were like this, and my 17hh TB was like this when I got him too. I didn't find it stiffness though, but a lack of balance in a large horse going round small corners. I corrected it by lots of lunging, so the horse learns to balance itself on the small circles without rider weight interfering, lots of hacking out, to build up fitness to carry the rider in straight lines, and by starting off schooling in a larger arena, and when riding in the small arena I did lots of walk work to start. I introduced lots of transitions from halt to walk and walk to halt, and then short bursts of trotting on the straight sides with lots of walk-trot-walk transitions, and then the start of lateral work in walk - lots of things that will help with balance at a slower pace.
For all three that I've had this initial problem, I have done this, and it has let them strengthen up and learn to balance, so that they were actually able to balance and handle the corners in the small ring. Bigger babies will find small schools more tricky to start, so give them time!