Leaning on the bit generally means the horse is going on the forehand and relying on you to balance. Get that hind end engaged! Trot poles are great for this.
Fab explanation below. Just to add that the action of the bit plays a big part too. The plate in the French link creates tongue pressure which brings the head down so can cause leaning issues for horses that like a lower carriage anyway. I would go back to a simple nutcracker action to bring the head back up as the action is on the bars. A hanging bit will create poll action which also lifts the head but is a stronger lifter than a nutcracker action. Maybe try a fulmer so you still have the cheek pieces but the bar pressure will help lift the head and the loose ring will just ease up your horse leaning on the bit quite as much, just a subtle difference :)
The reason why horse is leaning on a bit could potentially mean this is basically a way they catch their blance. This would mean your horse is a bit on a forehand or have weak neck muscles. To help your horse have a look at some exercises that will make your horse encourage its hind-quarters.
Going uphill and down hill on a cross country course, grid work or some small jumps/cavaletti should be a good solution to help your horse develop lightness in the forehand and enourage it to use its hind quarters. Lungeing with side-reins or passoa with some pole work and cavaletti will help your horse's neck become stronger.
Bit wise, you could have a play around and try your horse out in different snaffles like lose ring or d ring snaffle with one or french link or eggbutt snaffle. See if stearing your horse will still be as easy as when it was in a full cheek.