Red and Squiggy are the same in the way they do their body movements.
Red does: (if I do not do my job of keeping him correct)
--shoulder in or cut into the circle
--swing hip out and not engage it
--have head high and nose out
--sometimes ignore my direct rein and inside leg cue to bend
I see Squiggy doing the same thing to you in your video. So that's how they are the same.
Who pays for Squiggy? Whose truck/trailer do you use? If your parents are footing the bill for any of it, I can understand that they have a say in what you do because my parents (okay, well my dad) were the same exact way. Horses didn't go the vet unless they were dying, and they certainly didn't go to the trainer.
But if you are paying your own way for Squiggy, well then I'd take their "advice" with a grain of salt and go do what I know I need to do.
Does this roping guy who knows his stuff give one or two lessons? It's sometimes amazing how helpful it can be to have another person watch you and/or hop on your horse for themselves.
I know I've said it before, but I still really think you need to quit working her on barrels for the time being. Put the barrels away, and use your dirt area to work on TONS of circles, figure 8's, diagonals, roll backs, etc.
1) Never let her cut into a circle. If she does, she's gotta go around it again all the way until it's perfect.
2) Make sure you give her RELEASE. If she drops her head for you, let the pressure off the reins. If you never give her release, she'll learn to never give.
3) Always keep that hip in .... unless you you want to work on counter canter in a circle (quite a beneficial exercise) then you will force them to uncomfortably keep that hip out while on the wrong lead for a couple laps. You can then make a straight-away in a diagonal, or act like you are going down the long end of an arena, break down to the trot, and ask for the correct lead. They will love the opportunity to keep their hip inward after that!
4) Keep her collected. Right now, she's got a pretty hollow back. With the bending exercises in lots of circles, her head should naturally start to drop to find that release from YOU. When that happens, her gaits should slow down and her back should round. Bending leads to good things.
5) If she speeds up faster than you want her to go, bend her into a small circle so she has to slow down, then carry on. If she speeds up again, bend her into a small circle again. You may have to do this over and over again, but stick with it. She should stay at the pace and speed you ask her to stay at, without speeding up (or slowing down).
6) Your end goal should be self-carriage. She should start to always carry her head lower and she should start to always travel collected (and not hollowed out) because she knows the instant she doesn't, you are going to correct her. This is why I think you need to not think about barrels right now; simply get her riding better.
Curious .... have you ever had her in a shanked bit? I used to say that if you can't teach a horse something in a snaffle, then you shouldn't be doing it ...... until I took Red to that trainer. I had been trying to get him good in a snaffle. Wasn't working. She put a high port correction bit on him and almost instant results. Yes, harsher bit, one would say, but I do believe in using the bit that works. And the bit that works might be a stronger one. But if you can touch their mouth less in the stronger bit, rather than hauling on them in a snaffle, then maybe that "harsher" bit is actually softer than the snaffle. Because you are leaving them alone and applying less pressure. I only ride him in that high port bit on occasion, but it really keeps him soft.
Something to think about anyway.
Yes, I do see improvement in this video, but Squiggy still has a long way to go. Get her riding better over the next month with no barrels. You'll be happily surprised at how much better she'll turn and handle when you re-introduce it.
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It is not enough to know how to ride; one must know how to fall.