Don't tap. Ease. This is just like shooting: You don't pull the trigger, you squeeze it. Now, maybe this is a different "tap," but when I "tapped" the stick I would tap it, you know... just hit it lightly and let it spring back into place. Well heck. It ends up going a little left as it tries to recenter if you "tapped" it right. If you're pressing it right, then press less.
Good practice for turning is the Test Track with a car that can hit 190mph+. Keep doing laps, trying to stay in the same "lane" and not lose any speed on the turn.
Remember, it's analog for a reason: To give you a range of options. If all you do is make hard rights, then just use the D-Pad.