Holding the saddlehorn

Your excused.

Every beginner balances on the reins at some point. I'm sure I did, but my instructor took the steps to FIX the problem. We weren't allowed to canter until we could ride a whole lesson without stirrups.

No one actually intentionally hurts there horse, but it happens. If you catch your horses in the mouth every once in a while its fine. I do it, usually if the horse unexpectedly trips or they spook. Using your reins for balance will make it uncomfortable for your horse. Right now you can't use you body for balance so your going to use your reins or saddle horn.

Not "dictating" a horse is dangerous. They can't just be galloping up and down hills all day. That leads to dangerous behavior. I'm sure if Patty Gator had it her way she wouldn't want you "dictating" her out of her paddock, slapping a saddle on and galloping her up a hill. I don't dictate, I ask my horses. And they ALL enjoy their jobs. One mare loves reining the other is a fantastic trail horse. Plus I run a barn with 20 lesson horses. I know them all like the back of my hand too.
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I haven't been shooting you down, I'm trying you to tell you the God honest truth about many dangerous horse related issues you have been having.

You told me about Lily, I never asked.
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I wouldn't suggest holding the horn. Mane is better. Holding the horn will ricochet you off the horse, and I've almost had that happen once. Not fun.
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You should be checking the tightness of the girth before you get on. It should not be sliding around like that.

I could be wrong, but saddle fit might also be causing it. Doesn't matter how tight the girth is, the saddle will slip if its too large for the horse's back.

But that's a bigger conundrum
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^^Yep, too wide, too narrow, bars too flat or too angled. It will all cause a saddle to slip around regardless of cinch tightness.

All around OP this is a very dangerous situation, and if you really have dreams of being a show jumper you should find an experienced and certified coach to help you. You should not be galloping at all, and it seems like you have a lot to learn about horse behavior and their mentalities.
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Yeah, that's what my instructor always tells me - grab a handful of mane. Of course I can't grab the horn, because my saddle doesn't have one.

: p when I first started riding English, I would claw around for ANYTHING that I could get my hands under. Sometimes I still do when I terribly lose balance... Like when a horse trips.
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I don't know if your just telling us rare exciting spots of your riding, or if the lady you ride with is letting you run amuck on her horses. But, regardless I hope you hang around here and read the experience of others.

I'm all for having fun horseback, but you need to get some skills under your belt first.