In order to get real collection you need to tighten you reins until he gives his nose and release, this will teach him to carry his head and give to the bit. Once he does this (he may already do that much) but push him forward at the trot to an extended trot (posting trot) and drive him forward with your seat. All the while keeping his neck and top line level while his nose stays vertical. This manuever will build up endurance, balance, hindquarters and topline
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I'm assuming as this is posted in the english riding, its english riding, and I am not too sure if there are any differences between english/western.
Firstly, I think what you're trying to describe is a half halt. You don't just use the reins though. I have copied this from a previous thread I replied to as I feel that this is also what you require, OP. Headset and collection come from working from behind (not dragging on the front) and a relaxed back from the horse, half halts, leg and seat all used correctly from yourself.
Secondly, coming 'round' or on the bit comes from a mix of things, not just the bit.
You need to ensure your horse is warmed up correctly. When I warm up in walk and trot, I never go whole school, I have single loop serpentines, three loop serpentines, 20 m circles, change the rein through half the school, change out of the corner, 10m circles in walk. Anything, use your imagination.
If you struggle to get your horse to listen to your aids, and he doesn't work from behind, transitions. However, correct transitions. When you ask for walk to trot, or trot to walk/ halt it has to be on the dot, not teeper down in to it. When you think walk, he walk's. You have to prepare the horse with half halts on the outside rein, let him know something new is coming. You get him ready with your legs and seat then BAM, do it. Don't pull back on him. As you do more and practise more, it'll become easier.
Also position. When the horse is working correctly from behind, he'll start to swing in his back, and naturally drop his head, they do it in the field, free lunging, anything. Its comfy for them. You want to have him in an outline though... so, you need quiet quiet hands. I was always told my outside rein is my 'working' rein, and my inside rein is my 'direction' rein, direction as in left right, I'm combo with leg and seat, and also head position. If your horse resists, don't pull and fight, give and take. Your trainer should be able to explain this clearly to you. Keep sending the horse forwards, though.
Lower your hand position if you need to, and don't expect the horse to come up and neat and tidy straight away, it is very exhausting for a horse, roundess and suppleness does not mean he has to be up in your face. When he comes in to an outline, reward with a 'give' of the rein, I'm not saying throw it away, its a small action which makes all the difference in the world.