You can desensatize and get them to follow your orders, yes- but you cannot mature an animal, just like you can't mature a child before its ready. Babysteps. IMO horse training should be done slowly and with plenty of tiime for any animal to adjust itself to your way of thinking. A horse who understands is a horse that will not blow up.
So many horses are started as two year olds, and many of them have atleast a few bucking fits before they can settle down atall. This is why many trainers say to 'break them early', because they're still uncoordinated and dependant, making them 'safer.'. In all honestly though, any horse can be trained in its later years if you give it time to expand it's mental capacity to understand what you're asking of it.
I think of it like this. With a child, you as their parent have the job of slowly teaching them to take on chores and school work, in order to get them ready for the real world when they grow up. Under four, you generally only ask them to do simple things like behave in public, respect mommy and daddy, and clean up their toys when they're done. At around five they're asked to do a few small chores like shower themselves or put up their dirty dishes. At seven or eight they're asked to do a little more, and a little more. Same with school. You start out asking them to sit still for ten minutes at a time and gradually work up to expecting them to engage their mind the entire time that they're at school, up to seven or eight hours a day, and then learn to work hard every day as adults for 12-16 hours.
Its the same with horses. As really young babies, of say 0-18 months, you really dont expect them to do more than respect your authority (walking calmly on a lead, letting you handle any part of their body) and you want to just let them be babies. Sessions should be under ten minutes. As they mature a little and begin to grow, at 18-22 months or so, they start maturing a little more and you want them to tie, and do things like trailer or back up when you tell them to. But they're still immature and they're going to get tired easily. You want things to be all fun at this point. Play little games with them like the seven games of horsemanship, or maybe going on little 'adventure' walks with them. At two years old they're about at a young teenager's maturity. They're almost the size of an adult and look quite a bit like an adult, but they're not adults, and they aren't going to act like one. They'll be excitable and will be able to concentrate for a certain amount of time but they still need time to have fun. This is when you can introduce simple, nonstressful things like ground driving, bridling, and if they're mature enough, perhaps sitting a saddle on their back while you walk around with them may be possible when they're getting close to three. Right now you're just want to have a sturdy relationship with them, and they should trust you and be willing to try hard for you- but you have to allow for time to just be 'kids.'
At three they're young adults and so you can then begin introducing them to the work life. Teach them to be saddled and bridled quietly, yield to pressure on the ground and begin light lunging at a walk and trot. Get them into a 'working' mindset. It is generally safe to begin riding them at this time, as long as you take everything very slowly and are patient. They're still trying to grasp what you're wanting of them, and this is one of their most influential years. Teach them to walk and trot under saddle, and to adjust to your weight.
At four they're adults and should now be acting like adults. Harder riding such as at a canter and introducing ground poles and circles will be ok as long as you watch them for any signs of being overloaded. If they start to stress, go back a step and reteach.
But ofcourse, just like people- horses mature at different ages. Some might be ok to ride at a walk as an older two year old, some may not be mentally able to carry the responsibility of riding at even three or four. You just have to judge your horse by how well you know them.
I've seen two year olds who acted years older than they were, and I've seen four year olds who literally just spazzed out because they were on information overload. Just go easy, take your time. You'll appreciate it later.
Everyone in your life is meant to
be in your journey, but not all of
them are meant to stay till the end.