Barrel Help Please (Beginning barrel racer)

I have a pony that I consider an all-around pony because we do everything together
I use CA methods, specifically the lateral flexing thing he's got going on. (like flexing the neck from side to side).
Putts is really soft and sensitive to ride. He's not a great barrel pony, but we like it and he gets around the pattern.
When I'm barrel racing and we try to add speed, I'll turn his head around the barrel and cue him with my inside leg like I was told, but he'll turn his head and keep going straight for a while until his body catches up with his nose/neck and he turns around the barrel, making it a really wide barrel!!
He doesn't do it when we go slow (like at a walk or a slow trot), just when we canter it.
What are some things I should be doing to help this?
Thanks!

I, personally, cannot cue with just my inside leg. Inside leg gets pressure whereas the outside gets tapped so that he/she knows we're turning, not just bending around my leg or moving outward. All the horses I've ridden have been trained and ridden like this, as is the horse I'm training on barrels now. Though, while we can trot the pattern perfectly, she'll go wide around the barrels cantering. She just doesn't have the coordination to sit down around the barrels and keep going yet, so circles are going to be our friend for awhile.
Try doing canter circles and spiralling down to a smaller circle until you feel he might break into a trot and spiral back out before he does. Was the circle more than 20ft or so? If so, he needs more work at the canter on bending and collecting himself before you can run the barrel pattern and not have wide turns. When he can canter an 18-20ft circle both ways, then you can move back to the pattern. That's how I was taught anyways, and it's been working for me.
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I would use my inside leg only going into the barrel to get the arc, both legs in the middle of the turn, then outside leg to finish the turn.

I would go ahead and make him turn again until I get the turn I wanted, then let him come out of it. Also, on your slow work, you could go on ahead and walk to your rate point, push the hip up underneath him again, take a few more eps, push the hip in again, few more steps, hip in, then bring the shoulder around.

Do you have a video?

Personally I love the flexing however the mare I am riding now was taught it and rubber necks around the barrel even at the walk, I've found that the less pressure on the reins is the best with herjust the lightest pick up with her. It almost sounds like you are just pulling his head around and he is just turning his head (just assuming as I havent personally seen you ride :) ) but maybe try using more of your seat and just a light light lift with your reins as well as the circle exercise from above, those helped the gelding I was riding before as well.