I think your posting needs to be slower.. it looks like you're both chasing each other along in the trot. Slower posting, and encourage your horse to stretch. Have you tried combing the reins or riding on a loose rein?
Also.. your horse isn't balancing around those corners very well.. you should try sitting to the outside a little bit with your inside leg tapping his barrel over until he can learn to balance himself.
I would spend more time walking in your warm up. At least 8 minutes of solid walk work. Do circles at the walk, leg yields on and off the track a few times, change rein, serpentines. All without rein direction.. just leg.
Then when you pick up the trot, make sure you half halt to re-balance him and let him know what's coming. Timing is important.. the trot is a 2 beat gait.. both legs moving diagonally. So you need to ask for that trot with the right timing else he'll probably rush into it or take awhile to get sorted.
Then trot allll over the arena.. slowly. As slowly as you can. YOU determine speed.. not the horse. If the horse is just running around trotting.. and you're posting really quickly to keep up, that's only going to make him faster and more unbalanced and that's kind of what I saw.
Slow posting, longer rein. Stay on the rail 2 laps.. then serpentines, circles, change of direction.
Then once you get that nice trot going, then you can add some leg and ask your horse for a bigger trot but still post slowly. Your horse will just add more hind power.
Then once you have that.. same thing. Do figures. Mix it up.. then break into a canter for a few strides.. back to trot.. canter... back to trot, then canter and stay there for a longer period of time.
Warm up is that.. warming up the horse. If you just speed through the warmup.. then your horse isn't going to be at his best, you won't be warmed up either and will get tired, stiff, and possibly more sore, and the work won't look "together."
I say this all with a smile on my face, as I am just trying to help.
I can't really see anything going on with your seat as it's so far away, but I guarantee if you warm up properly, your seat will definitely improve. That and walk without your stirrups (I'd advise only walk now until that trot becomes a lot slower and more put together.)
"Strength is the ability to use a muscle without tension"