Jessica actually rides him very well, but you're definitely right that she needs to release. Rupert is 20x better when you give him his head.
He isn't green, actually. He's about 12 and an x-show horse, actually used to do some fox hunting..
He needs the martin gale because he pulls his head up/down a lot. I might start trying him without the martin gale, but he does need it sometimes..
He has breating problems and this was a pretty long course, so he wasn't too pleased with having to run around and around. I don't show him[ this was a show ] and when I ride him, we take plenty of breaks, so that's pretty much solved.
He does lack a lot of confidence though, he bought him after a while out of work.. But he's getting way better and is really excited to see the fences most of the time now.
I have him with a trainer twice a week, and we've only been doing small things, but I'm going to ask about getting him schooled a bit more often. .
Honestly from what I can see it looks like he's pretty sloppy as far as jumping goes. His hind end is very uneven, showing me that he's not sure how to use himself and as a result he's throwing his upper body over the fences. I'm not at ALL a fan of putting a martingale on a green horse OF, especially not a standing one, so my first recommendation is to remove that. Next, he needs a more confident rider to help him learn to use himself and gain confidence over fences.
I know you said that isn't you riding, however it's important to point out that the rider ABSOLUTELY affects how the horse will jump. In this case the rider is lacking a release and holding the horse in the face. As a result, the horse's ears are pinned, his facial expression is a rather unhappy one, and he's trying to both safely jump while at the same time not being allowed to properly use himself over fences.
Not sure who the rider is, but if he was my horse, they would be lessoning with a trainer on a different horse while this horse learns confidence and balance over small fences in the form of gymnastic exercises that include both trot poles as well as small fences no higher than 2' of varying widths and as he gets more experienced, distances.
In addition, based on what you said about the horse tending to overjump, I have a feeling that he's both lacking in hind end as well as perhaps avoiding the rider (if anyone else rides him like the one pictured OR if the rider pictured rides him regularly, he's learned that he needs to avoid the fence as well as avoid the strong pressure on his face - making jumping a not so great experience). Be careful as with continued rides like this, it's likely a matter of time before he starts to refuse as right now this jump (in the picture) is an example of how to teach a horse to dislike jumping - with the common outcome being refusals since actually jumping as above results in him being held in the mouth/caught in the mouth.
Looks to me overall like the horse has a lot of work to do on his form and the rider needs a quiet packer so that she can work on her own position and learn to ride with a much more secure leg/seat and develop an independent hand over the fences so as to not interfere with her horse. The rider would be best suited for a well school jumping lesson horse with a trainer, and the horse looks to need some rides with an advanced/professional rider to teach him how to engage the hind end first on the flat, and then tighten up his form over fences.