Much easier said than done, and to be honest, I hate when people give advice like that. I've gotten it from track instructors too, and I don't think that's overly wise, especially from a track day instuctor. Some people may just have a phobia that takes them a lot longer to get over. And bear in mind, that you can read everything about the logistics involved in turning or what ever, and there is literally NO way to prepare or "try out" it unless you are actually doing it. Everyone says lean it more, do this, do that. Well if you have a phobia about doing something and are skittish about committing to it because there is no way to prepare, that WILL take some time to "get over it".my advise would be just to get over it. Thats the best way I have gotten over my fear.
To the original poster, I was the exact same way. I was so better at left turns than rights, then I had a low side on the right side, and I still was shy on right turns. 3yrs later, I have a low side on the left side, and now I can't turn left worth a shit. I am FAR better at right turns now than lefts. I think there may be something to the right handed/left brained thing, but I tend to think that some people might be more afraid of right turns because that's the side of the throttle. Maybe they're subconsciously afraid of giving it too much gas. I'm far from an expert but that could be a factor for some people. Just ride your ride and don't fuss too much. I've reached the point that I'm just giving up on the things I want to do with riding (trackdays, taking corners faster, getting the knee down, etc..) because I've been told I'm too slow, that I can't do it etc. Fuck em, I'm happy to ride anyway and that's what I'm going to do.
edit: I am right handed too, but I now can almost get my knee down on rights, maybe 8" off the deck. Left I'm practically upright on turns sometimes.