When a horse refuses to do something like this, the first thing you always want to rule out is pain --- especially since he is an older horse.
Have you had an equine lameness vet check him? (Not just your local "all animals" vet.)
Have you had him checked by a chiropractor?
Does he need dental work?
Does your saddle fit properly?
Is the bit pinching him and/or fit properly?
Will he turn right out in the pasture on his own, or when lunging him without a rider? (Or does he only do it when you ride him?)
If you can rule out a pain issue, then you can try to begin to fix the issue.
Start with baby steps. Start with ground work. Get him to bend and flex in both directions. Get him to lunge and turn in both directions on the ground. "Re-introduce" the bit while you are on the ground, to teach him to direct rein to the right and left. Get him turning to the right consistently on the ground first, and then try it from the saddle.
Be patient. Be consistent. Do NOT stop asking him to turn to the right, until you at least get a small correct response, and then praise. But you need to release the pressure immeidately when he does something right.
∞•*˚ Βгįťţαňγ ˚*•∞
It is not enough to know how to ride; one must know how to fall.