Eye8 is right, check out your battery cables to make sure they are tight and making good contact. Rule that out first. Then turn to your battery/alternator since these are very easy to test and you're suspecting them.
If you can, bring the battery into a store to have it tested, along with your alternator - the light doesn't always come on when there's an issue with it. If you can't do either, or just want to diagnose ASAP, start with the alternator.
If you can't bench test the alternator then test it this way:
- Start your car
- Unplug the negative battery terminal
- If the car stays on, the alternator is good. If it turns off, your alternator needs replacing and, most likely, your battery does too.
- Reconnect the negative terminal
- Test your battery, regardless of the outcome
You stated there's a clicking sound. I'm assuming this is after turning the key. This can be from your battery or starter.
Bad starts and turning off while driving points to your alternator my friend. Of course, can be a million other things but just starting with basics.
Funny you're having this issue on right turns (maybe just a coincidence), but I had a 91 GT-S that blew my cluster fuse on a sharp fast right turn once!