Go right back to basics.
A horse cannot swing through the back, if it is not relaxed, infront of the leg, and submissive to the bit. This does not mean pulling the bit side to side to 'soften' the horse's head, then kick it a bit to go forward. This will give you a false frame - the horse will build incorrect muscle - most noticably under muscle on the neck. It will also tend to back off the contact and hold its neck set in one position. In a dressage test, you may be asked to give the reins and have the horse stretch long and low for a full circle - if the horse is trained to 'soften its head' by having the bit wiggled, you won't get a true stretch in this movement, because the horse is NOT using it's back.
So, basics first. Make sure your aids are reacted to positively, every time. He needs to be in front of your leg, with your seat and not running. Put your leg on, if he doesn't go straight away, give him a flick behind your leg with a dressage whip. When you're moving, if he tries to pull on you and barge through your seat, halt. Then start again.
It is tempting to ask the horse to work 'long and long' straight away. But this is actually a very strenuous exercise for a horse to perform correctly - and if it's not correct, it's useless because the horse will be on it's front legs.
I like to ride a green horse on a steady contact, with reasonably long reins that give it something to work into.
Ride MILLIONS of transitions, changes of rein and transitions within the pace. I can't stress enough how important it is to stay on curved figures and continue to change the work. Straight lines are for testing, curved lines are for training.
Don't get stuck in the rut of riding circle after circle, not changing anything. Ride no more than 1/2 a 20m circle without changing something. Either slow the tempo, speed it up, change pace, put in a smaller circle in the opposite direction - be creative. The more you change things up, the better the horse's balance will be. He'll NEED to start shifting weight to his hind legs to be able to balance.
Try introducing some leg yields in and out of a 20m circle. Leg yield out, then ride a 10m circle in the opposite direction, then leg yield back in.
The more work you do, juggling the horse between your aids, the stronger, more supple and more swinging he will become.
Always keep your contact steady, there is no shame in holding onto the saddle blanket to help you if you don't have perfect hands. The hands should never interfere with the work, they are only there to add flexion, provide a contact and very occassionally give a correction. I will often ride with the velcro straps on my saddle cloth in my hands, it is a great reminder to keep your hands quiet!
After all of this work, if you're riding well, your horse will be starting to stretch into the bridle, stretch out from the wither and be more active in the hind legs.
You can then start refining half halts, which will transfer the horse's weight back to its hind legs, then ride forward again.
If you don't have access to a trainer, putting a video up here would be very useful, so we can see exactly what's going on and give advice more specific to your case - every horse and rider is different so all I have given you is generic blanket advice.