I’ve been watching you for a little while but unfortunately have had a lot on my plate and no time to write anything. I watched your latest video to give you the fairest assessment I can.
One of the first things I noticed in most of your videos is you’re tight in your hips and thighs, your hips are contracted to suck your femur and knees to the saddle (this I think is why your leg was swinging so much), not that it was bad but it can be quieter by you allowing yourself to relax into your seat and move with the horse. Don’t force your seat to move, just follow. You shove a little in the walk and I think it is because he is not forward enough in his walk and so you compensate with your seat to try to get him to step up but he’s not responding. I would ask for a bigger walk with your leg and reinforce with your whip if he doesn’t respond to a bigger aid (minimal use of whip, whip is just reinforcement to help the horse understand what is expected). Or you can alternate the use of each leg with the swing of his barrel left leg, right leg with the swing to accentuate his walk and insist he take a bigger step and listen. At this point communicate with leg first and let your hips follow. This will be more productive. Anytime you apply an aid, expect a reaction. If he doesn’t react, reinforce your expected reaction but be fair. If he isn’t getting something take a step back go back to walk and see where the miscommunication began.
I think some basic work in turns on the forehand and leg yielding will help you get the connection of his inside hind leg into your outside rein and get him to really round around your inside leg. You may have to accentuate a bit to get him to mold around your leg and step under with his inside hind leg and allow himself to be in the outside rein. It will really help you to keep your elbows at your side with your shoulders down and back, when you put your elbows so far in front of you it’s hard to keep your position and stay connected through your whole body (you should feel the connection in your whole arms and into your shoulders and back). I’d also keep your outside hand lower than your inside one for a little bit to help you control the outside shoulder more effectively (really make sure the outside rein is low by his wither in transitions, this will really help you). Having the outside rein lower will help you so he doesn’t fall out through the outside shoulder when you’re asking him to mold around your inside leg and step under with his hind leg, in the sit phase of post or downward phase sitting is when you half halt on the outside rein and keep the connection in it no matter the fuss (occasionally give the inside rein completely to make him seek the outside) and emphasize the bend comes from your inside leg and not your inside hand. Sometimes it helps to think when you want to use your inside reins you use your inside leg. You may also need to help guide his shoulder in with your outside knee, inside shoulder forward or think shoulder fore if his shoulder is going out while you're trying to get him to use his inside hind leg. You may have to exaggerate how much bend you want or sometimes have a little feel of the inside rein (simply lifting the inside rein up or squeezing your fingers and releasing) to remind him to bend his whole body but do not drop the connection of the outside rein and do not lift the outside rein up, keep it at his wither. If he doesn’t understand this at canter, go to trot and if he doesn’t understand it at trot go to walk and accentuate what your inside leg and outside rein mean. To do this you can do turn on forehand squares leg yielding to each corner of the square than accentuate the turn on the forehand and once he gets it let him out.
In your transitions you really need to keep your connection, think elbows at my sides, shoulder down and back, outside rein low. Leg yield out into the transition hold a second as he starts to transition give both reins and send the horse to the bridle in your downward. A few times you let your reins to long and lost the connection, start long and low with feeling the horse pushing towards your hand but not leaning (add leg if he leans and put the reins forward a second) and say no you have to carry yourself, having a rein that is too long won’t engage his back. You still need a connection in long and low. This is the same for both upward and downward transitions, you should not hold back in a downward transition but sending them into your hand and them listening to your position and body. To transition down I use my back and seat and close my thigh for a moment, hold the connection for a second by closing my fingers, leg yield 1 step and give the reins expecting the horse to reach into the bridle
Be definite in your circle shape and size, make it consistent. Put him on a 20m circle. Sometimes put him on a 10-12m and leg yield out to a 20 and leg yield back in to help you with the bend. Expect him to bend around your inside leg and feel him reach into the outside rein. Ideally you should be able to throw away your inside rein and he'll still be round and balanced. Tips for leg yield, step a little into the stirrup of the direction you are traveling, look to your destination slightly and add leg as you feel his rib cage push into your leg. Add whip if he doesn’t respond to your leg.
I’ll give you a visual. I wish I could put you on my baby horse to demonstrate. If I ask my horse for a leg yield I put him in shoulder fore and accentuate his haunches travel almost more than his front end. In my head if I leg yield from centerline to quarter line his haunches shoulder touch quarter line before his shoulder. My goal in the leg yield is not to get from point to point but that he steps under himself and uses his body correctly. I can feel a big difference in his back and can really feel him stretch into the bridle after a leg yield. I can feel his legs are that much more engaged as he connects his whole body with mine. In shoulder in I can completely throw away my inside rein and he will still be in the connection, still in shoulder in because it is my inside leg that becomes the new inside rein and tells him to round through his rib cage and my outside rein half halt (again just closing my fingers and releasing) catches the outside hind leg and keeps him in balance. He knows the degree of bend from my position and how much outside rein I use.
I know this was long but I hope some of it made sense, I try to be detailed because I don't know what will make sense or not and I don't know how you think or understand material but I hope this was at least helpful.