For JC Racinet, the dominant use of medium trot is one of the aberration of modern training.
Indeed, too many people consider the medium trot to be the normal trot their horse should give them. But before asking your horse to go in a big trot, you should make sure he is using his back properly.
Especially with young horses, this is a difficult gait. The horse has already to use a lot of energy upward, to get his topline to sustain his whole body and your weight. The more muscled he'll get, the more energy he'll have at his disposal, "stored" under your seat. It's only when you feel that you have enough gas under the saddle that you can ask a bigger trot.
If too early, you ask the medium trot as you're asking your horse to raise his back, then you'd be asking for two different and difficult things at the same time : the lengtening of the trot and fastening of the rythm - forward energy and the raising of the back - upward energy. And because it's more easy and urging to give in to the leg order, the horse will hollow his back and rush forward. You'll get the medium trot but you'll lose the back.
The impulsion is not something you ask, it is something you gather, a situation you create.
And it takes some time until your horse is feeling as confortable and powerful in his trot so that you feel you have extra energy to use in a bigger trot.

Time is patience and time is money. But going slowly is the best way to go fast, as classical master once said (please help if you know who!). Many people try to go over each step without making sure the horse is already performing the first one with no more physical or mental tension.
And most of all, the desired frame, the way to collection is very long and shall not be hurried. Don't force a young horse into a frame he cannot sustain because
- his body is not ready
- his brain is already overwhelmed by all this new job he has to do
Don't add pain and stress, and don't get despaired if during the first months, the head move up and down. Reward when the horse stops fighting, disaprove verbally when he fights and get tensed ; but don't force the frame with a steady and heavy contact.
On the lunge line, no side reins or anything, just let him find by himself that everything is better when he puts his head low. And he will ! They understand by themselves, always... Sometimes it takes longer, just be patient. You'll protect the body of your horse from any tension created by side reins that overtense the top line in a contracted unfreely manner. And eventually... save you the time of tightning all this ropes and buckles !