It's unlikely to be carpal tunnel, though that's a possibility.
A major nerve in the hand, feeding the fingers, runs across the palm, right where one tends to lean on the bars. Constant pressure on this nerve irritates it and causes the numbness and tingling that cyclists often experience. I'm not sure how much irritation it takes to cause actual damage, but I suspect in most cases the situation becomes intolerable before serious damage is done.
In addition, pressure and the constant cramped grip that especially new cyclists tend to use cuts off blood flow in the hand. There are several (relatively) major blood vessels that run across the palm.
The solution, of course, is to relieve the pressure on the palm. Padded gloves help, but beyond a modest amount padding can be counter-productive since it spreads the pressure out over a wider area, causing a wider area to be deprived of blood and numbing out more area such that a minor change in hand position doesn't achieve a "fresh" grip.
I once had some special gloves that were really good -- they had "Sorbothane" padding (a sort of rubber), but, more than that, the padding was in the form of narrow ribs, so that they sort of "massaged" the palms as you rode. But they wore out and I've never found replacements. (I've often thought someone should make bar grips with similar ribs, but I've never seen anything.)
Otherwise, the best solution is to relieve the pressure on your palms by adjusting your riding position. Raise the handlebar and move it closer to you (more to the rear). And maybe move your seat forward a bit.
The other thing to do, of course, is to change hand positions frequently. Traditional drop bars give you several good positions for riding, and with straight bars you can get bar extensions to give you alternate positions.
Added: One problem I've noted with bikes these days is that they're built to "look mean" on the sales floor, with the handlebar unrealistically (for most people) low. And with the threadless headsets raising the bar beyond a half-inch or so is apt to require changing out parts, something people often don't know is even a possibility, even if they aren't reluctant to do it for various emotional and financial (and macho) reasons.
Added: I once ran some back-of-the-envelope calculations to see how much pressure the body was placing on the hands, vs blood pressure, and what I worked out is that upper body mass and leg/back strength (which affects how much weight relief the legs offer) are critical factors. When someone is over roughly 200 pounds with poor strength (which I have due to polio) then it's impossible to spread the weight out enough with padding to keep from cutting off blood flow. If you're lighter or with better leg/back strength then you're probably wondering what all the fuss is about.