Am I the only one bashing X button to run and he doesn't move faster at all? I've eaten well so the stamina and health are full and holding X or tapping it won't make it run. I can't even pass the first mission that I have to follow a group because they're running faster.
There are some areas like your camp where movement speed is restricted to a rather slow walk. I suspect you were in such an area. Tapping X should work usually, both on foot and on your horse.
Play in first person. Then you'll default to running with the left stick pushed fully forward. And it looks better.
Press L3 to run X is to crouch