Watch YOUR body position through the whole run.
You have your hand on the horn the entire time. You are sitting down in the saddle. You look like you are out for a trail ride.
If you have no enthusiasm in your body language for the barrel race, why would your horse?
Here's a few examples of how a barrel horse should be ridden.
I think Sherry Cervi is an amazing, quiet rider. Yet, watch how she is subtly hussling her horse Stingray. Her body position is forward, she's driving her with her legs, and she's hustling her on the way home.
Or Brittany Pozzi riding her horse Duke. Brittany is a small gal, and Duke is a big horse. But she's pushing him in-between each barrel.
And Lindsey Sears and Martha, who have a very unique "wrap around" turning style. But Lindsey is push, push, pushing her through the run.
I would not use a whip, or spurs, or over on and under on your horse. I would try to fix it with YOUR RIDING first.
Also, does your horse know how to run? It seems like a silly question, but have you ever just breezed him in a wide open field or track? And let him stretch out and RUN? I think it is very important that a horse learns how to run away from the barrel pattern, because they will figure it out sooner or later in the arena with fences.
How long has he been on the barrels? Sometimes it just takes time until the light bulb comes on.
Also, I would give him more room around the barrels in your pocket and turn. This will encourage him to keep his speed around the barrel. As you already know, you turned him way too tight and nearly stopped him on the first barrel. Go ahead and allow him to take it a little wider and keep speed. It is much easier to get a horse to tighten their turns with speed, than it is to get them off the barrel.
Get your hand OFF the horn in-between the barrels. Sit up out of your seat and drive him foward, and deep into the pocket.