We don’t know what’s wrong because his legs look good and all, but maybe he just don’t like this side? Does it happen to other horses too?
Just like you may be right-handed or left-handed, horses are too. They almost always have a side they prefer better. Most horses are left-handed which means they like the left lead better (and of course the term you are searching for is "LEAD"). When a horse is on the left lead, the right hind foot hits the ground first. Followed by the left hind and the right front together. And then lastly the left front foot.
It is called a "lead" because the horse's left front foot is LEADING ahead of the other one. That's why it is called the left lead. And yes, works the best if you want to ask the horse to turn a circle to the left. Their feet are in the proper position for them to execute a turn appropriately.
Whatever, the horse I’m training with is good on the right side, but when galloping to the left, he almost always start galloping on the wrong foot and I need to slow it down for the horse to start galloping the right way.
So it sounds like your horse actually prefer his right lead, which is okay. You'll just need to work hard on the other side to help him strengthen it.
Now, does you horse have any possible soundness or lameness problems that might inhibit him from wanting to go into the left lead? If pain is a possibility, then you should take him to a good lameness vet first to have his pain problem addressed. If something hurts him when he tries to gallop on the left lead, of course he is going to want to avoid it.
The key for getting a horse to take off on the correct lead you are asking for is to make sure their body is ready. They need to be collected enough to they can reach that hind leg under their belly to push off into the gallop. And they need to have their HIPS and SHOULDERS in the correct location to get the correct hind foot in the starting position.
--For a left lead, you want to push your horse's hips to the left. This will cause them to "step under" themselves with the right hind foot, so that it can be the first footfall.
--For a right lead, you want to push your horse's hips to the right.
For the shoulders, you just mostly want to keep them straight and out of the way. Don't let them lean or drop a shoulder.
Honestly, I care less about what their head/neck/nose is doing -- so long as the HIP and SHOULDERS are in the correct position.