Let's say you want the horse to turn to the right. So you would lay the left rein on the horses neck and give pressure with your left leg.
That is the simple idea, but let's make sure you are not accidentally giving a wrong cue somewhere else.
Keeping the same example of turning to the right, you should make sure you are NOT making any contact with your right leg. You want to "keep the inside door open" so your horse has a direction in which they are free to move.
You also should make sure that you are not trying so hard to lay the left rein on the horses neck that you are accidentally causing bit action on the left side. If that happens, your neck rein is telling the horse to turn right, but the action on the left side of the bit is telling the horse to turn left. This is really confusing for the horse! So make sure you are keeping a loose rein when you neck rein.
Another thing that may help in doing a right turn, is you can bring your right leg (the leg that is applying pressure because the horse should move away from pressure) slightly forward toward the horse's shoulder. This just encourages the horse to turn their front end only, and keep their weight and balance on their hind legs. Essentially, this is a rollback if you would turn 180 degrees where the horse crosses their front legs over one another while keeping the right hind leg planted (for a right turn).
If the horse is not responding to your neck rein to turn right, and you are holding the reins in your left hand, you can then use your right hand to pick up on the inside rein and give a direct cue to the bit to reinforce what you are asking the horse to do (turn right).
If the horse wants to drift into the center, so be it! Let's go to the center and WORK. Do anything you want in the middle of the arena but keep that horse moving. Do circles, serpentines, backing, and anything as long as his feet are moving. Then go back to the rail and continue what you were doing at a nice easy pace of walking (and relaxation). If he still wants to go to the center, then he has enough energy to WORK some more! Make the horse work where they want to go and soon enough, they will learn it is much easier to just stay where you put them (on the rail, in this case).
Another thing you can do is turn circles toward the fence. So if you are walking in a counter-clockwise circle around the area and your horse is not really listening to your right neck rein cue to stay on the fence, do a full 360 degree right turn into the fence to say "hey! We need to stay on the fence". Then continue walking. If he still wants to drift into the middle of the area, do it again. Eventually, he'll figure out that its just easier to just walk straight along the rail instead of having to do all these turns.
It just takes some FIRM consistency!!
A lady told me that when she was teaching her horse to neck reign (I've never tried it myself), she hooked the right reign to the left side of the bit and the left reign to the right side of the bit. Sorry if that's confusing, but that's the way she told me. That way when she would make the exagerrated movement to turn the horse left, the right reign would lay against his neck but tug a little on the left side of the bit.
This makes no sense and is giving the horse 2 completely opposite commands.
Like I explained above, laying the left rein on the horses neck (with no bit contact from the left rein) and reinforcing the cue with a direct rein pull directly on the right rein is the proper way.
By your explaination of criss-crossing the reins, you are doing the opposite and telling the horse to go one way with the bit, but telling him to go the opposite way with the neck rein cue. Super confusing for the horse and for the rider.