Welcome to the forum!
Its strange he will walk but not trot or canter past them. My first instinct is that he's testing you, or has learnt acting scared will get him out of work. How long have you had him? If you haven't had him long he may be testing or is insecure, perhaps one of first few times he went through something happened to convince himspooking at a jump = stopping trot/canter?
New or not, you could try lunging him in the field around, even over, some of the jumps. It gets him trotting/cantering around the jumps, and lunging and ground work is good for building relationships and enforcing your leadership. (If you haven't lunged him before, lung him somewhere safe to make sure he knows what he's doing- some horses never get taught lunging!)
And take him for 'adventures' to new places on hacks and that. If he sees something spooky, ride straight past it. There is a great thread in the training section of this forum on how to train spook free horses This is how we train a fearless trail horse!
. Read this, it will help you when schooling around the jumps to try and reduce spooking at them too. When my horse spooks, I ride her straight past it- she should accept my leadership and believe me when I say its ok and nothing to look at.
If he isn't new to you and you have a sound human-boss, horse-follower relationship, can you rule out pain? The trot and canter are more lively paces so the saddle moves around more-does it properly fit him, has he changed shape since it was last fitted? Does he have any sorness along his spine? Run your fingers down either side and see if he squirms or acts like he's in pain. Have you changed bit and it could be less suitable for his mouth? Is there anything that could be causing pain? Get a vet out to check him over if you think he may be in pain, or a saddler to refit his saddle.
As for the refusing, could he be feeling over faced? Did he do 'this stuff' with you or a previous owner? Is he used to jumping that height, that style? Has he built his confidence up over time, and does he have confidence in you when you jump? My horse is only just learning to jump and when she feels insecure about a jump she thinks about refusing or ducking out and needs my firm guidence to tell her its ok. I'm basically saying does he trust you to get him over that jump?
It may also be useful to get an instructor or experienced rider form the barn to come out and watch you ride and give you a lesson/guidence. They will be able to give you an outside view on whats happening and hopefully give you guidence to resolve this issue. Good luck!