Lots of flatwork. You need your horse responsive enough to your leg so that when you tell it to move out, then it moves out. Transitions, turns on the forehand, shoulder in/out etc., will all help to get the horse listening to your leg. After that, you can work on the turns. Plenty of circles, serpentines, figures of 8, and loops with help practice the basics, using your legs and reins. Half halts just before a corner, and before your turn helps to steady the horse and prepare it for a turn. After that, you can work up to barrels and even weaving.
Inside leg to outside rein. Inside leg provides a pivot point and helps to keep the horse out, outside leg pushes the horse around and stops the hind quarters from swinging out. Inside rein holds the flexion steady, outside rein controls the speed. Remember, on a turn, you aim to bend the whole horse around your leg, not just turn its head. The horse should be straight, even on a turn, with the hind legs following the forelegs.
A balanced and strong horse on the flat will allow for sharper and safer turns when jumping. Flatwork is boring to a lot of jumpers, but except for the actual jump, everything in between is flatwork and a good foundation of flatwork is essential for successful jumping. It might be worth contacting a good dressage instructor to help.
You need to do a lot of flat work, to improve suppleness of your horse, you will work a lot on circles. For example if your horse horse falls towards the right shoulder: on left-hand circles, keep a firm outside rein and ride with a straighter neck than your horse offers. Ensure your own weight remains to the inside of the saddle, not allowing yourself to be pushed outward.