Check your armrest position. Seems a simple thing, but IMHO it's the key to smooth flying on the Airbus.
1. Your whole forearm needs to be supported - straight line from elbow to middle of sidestick. Make sure your wrist isn't tilted up or down as you hold the stick. Get this wrong and it's really hard to fly smoothly, and you're at risk of making inadvertent pitch inputs during turns, and vice-versa.
2. Gentle grip. I fly with three fingers, and usually use just two fingers for gentle manoeuvres. It's an incredibly responsive aircraft - small, smooth, input, wait until you get what you want, and release.
3. In normal or alternate law, no backpressure is required in level turns (up to 33 degrees). If you're doing this instinctively, it might be why you're struggling. If you're not doing it deliberately (see "1") then it's *definitely* the reason.
Enjoy. Even if you just fly level turns for 10 minutes in your next sim, you'll see that it's a rewarding aircraft to hand-fly.