Side reins and draw reins might help to develop the muscles a horse would use when on the bit, but they don't necessarily help with the training of the horse as it generally doesn't make it easier to bring them onto the bit when the horse isn't using them. When I had this problem with my horse, I didn't realize the important of the emphasis on using leg. It's really important to use a very strong leg as opposed to a strong hand, and every horse has an individual balance of both, but using too strong a hand generally only leads to a fight.
When I had this problem with my trainer, we would start with the walk. Gently playing with the bit and encouraging the horse to soften his jaw and keeping the walk with a good quality by using a strong leg.
Only when the walk was a good quality, did we move into the trot- which would by default be of a decent quality after coming out of a good quality walk. Essentially, just make sure each gate is of a good quality before moving onto the next one, and this will make your life a little easier.
It's important to not get frustrated and be aggressive with your hands, and if you still have problems with bringing your horse onto the bit, make sure that you are working on a level surface, and as a last resort maybe try a slightly stronger bit. Take your time with this and it can take a very long time, and you'll find that the more your horse works in a frame the easier it will be as he has the muscles and understands what you're asking.
I'd recommend lateral exercises and lots of transitions every 5-8 strides. this will engage the hind end, encourage lifting the back and as the horse reaches over the shoulder to create a powerful stride the head will come down and then back up.
some good lateral exercises are spirals (the harder the horse the smaller the circles), side passes and outside leg to inside rein.
another good exercise is going around 3 barrels 6 feet apart.
doing all of these things you will get your horse nice and round around your leg without having to use harsh bits.