There is more to 'turning' than just changing direction! It is a lot more complicated than that if you want to teach a horse to turn correctly and not just change directions of travel.
First, ALL of the horse has to be TRYING to turn. RESISTANCE is what is the actual problem and it can be manifested in many ways. You want the horse to TRY to turn. This means that his entire body, from nose to tail must be TRYING to turn. Every horse must be taught to 'follow its nose'. A horse MUST follow its shoulder, but it has to be TAUGHT that its shoulder must follow its nose. This is usually referred to as 'being between the reins and between the rider's legs'. That just means that the horse is not 'pulling' or 'pushing' against either rein or either leg. It means that the horse is happy staying right where the rider want it to. This is a horse showing NO RESISTANCE!
Every horse has 5 body parts. Head, neck, shoulders, ribs and hips. They must all be taught to let you control all 5 of these parts 'without resistance' if you want them to turn correctly.
1) Head -- The head is the easiest part to control. Teaching lateral flexion and teaching a horse to bring its head around enough that you can see the corner of its inside eye is all you need to get this done. This is easy. Now, all you have to do is get the horse to 'follow its nose' and go where you want it too. When a horse is turning or working on a circle, the rider should be able to just barely see the corner of the horse's inside eye. Any more bend than that and you have a horse that is 'over-bending' which is a bigger problem than having a stiff horse.
2) Neck -- The neck is also pretty easy to get to bend. The problem is that it should have the amount of bend in it that you want and need and no more. If a horse is going to follow its nose correctly, it cannot 'over-bend' or 'rubber-neck'. It should also not be 'stiff' and not giving its head and neck and should certainly not give its head and neck and then keep going forward or the opposite direction. [This of a horse that is bolts and running off while its head and neck is bent way around to one side.]
3) Shoulders -- The shoulders must move laterally when you want them to. It is easy to get a horse to 'yield' its hind quarters, It is much more difficult to get a horse to yield its shoulders. Most people like to teach a horse to yield its shoulders on the ground before asking it to do this under saddle but this is not necessary. But, one way or another, a horse has to be taught to place its shoulders where the rider wants them to be. This usually means that the rider must 'pull less' and 'push more' so that the horse does not learn to 'over-bend' and push its outside shoulder out. Any time a horse resists by over-bending, it will 'drift' shoulder first into the outside shoulder. The rider MUST stop pulling and instead, must push the horse's shoulders over until it is 'following the its nose. If the horse has developed a habit of doing this, it may take tapping on the horse's outside shoulder to get it over the bad habit of over-bending.
4) Ribs -- If a horse is going to learn to travel correctly, it must mend through its entire body. This means that its rib section -- that part of its body that is between its shoulders and hips. The rib section usually does not 'over-bend', but in some horse horses it can be very stiff and resistant. This is particularly true of heavily muscled horses that are very wide and thick. A horse must learn to do 'leg yielding' exercises to help them learn to bend and 'yield' to leg pressure.
5) Hips -- A horse must also be taught to place its hips where the rider wants them to be. Many horses, especially heavily made 'thick' horses, have a problem keeping their hips in line with where they should be when they turn or make circles. This is why a lot of stiff horses cross fire and cannot keep their hind quarters where they should be. The best way to see if a horse is keeping its hind quarters where they should be, is to have someone on the ground watch it. On an ordinary turn or circle, the horse's hind feet should be tracking in direct line with where the front feet are tracking. On a fairly small circle, a person watching should not only be able to see the hind feet following directly the tracks of the front feet, but the outside corner of a big square saddle pad should stick out showing the horse is bending from its nose to its tail.
This ended up longer than I intended, but I hope you have been able to wade through it.