We definitely do have set beliefs about medieval horses, mostly incorrect. Just because we see illustrations of medieval lords riding what look like ponies too small for them, we think it must be the fault of the illustrator. But no, for journeys they really did have small trotting horses that could keep going on and on and on, with a swift steady gait that was comfortable for both horse and rider. To our modern eyes it looks a little silly, but they knew what they were doing.
It’s the same with similar illustrations that show them riding with straight legs, especially knights in armour etc. But that was the way they rode, and very sensible it was too, enabling them to “stand” in the stirrups with great ease.
As you can no doubt tell, I’m not a rider, and can only describe what I see, but if you go to this video you’ll see someone who definitely knows it all telling you about medieval horses. Thoroughly recommended.