Fear not. It is common for people that are new to Bersas to shoot low.
And funny as it may sound it is you. and it is the gun. But at the same time it is neither.
Ok. There is nothing wrong with the gun. And you are shooting just the same as you do with any other gun. Good your half way there.
Bersas are great guns and like ALL manufacturers they do things a little different than the next guy.
Three things are big for me in shooting a pistol properly. Proper grip, which you obviously have down since you shoot other guns fine. Proper sight alignment, which may be your problem and proper trigger pull which is the normal problem most people have with Bersas.
Trigger pull is an issue many people have, many people put their index finger in/on the trigger at the first knuckle, it should be the center of the pad (where your finger print is) of the index finger. This video explains grip and trigger very well.
The sight is the third item on my list. All guns can be different. Some use POA Point of Aim and some use a Six O Clock sight.
The above link is to a picture of the two sight pictures. You will need to try both of these to see which one your Bersa is set to. Note that the Bersa 380 thunder uses a different sight picture than the Thunder UC Pro and the HC Pro.
Back to the trigger. Once you have the trigger down, the next thing you need to know about the Bersa trigger is that the trigger is hinged a little forward on the frame than most pistols which until you are aware of this and correct your pull for it will cause your pull to bring the barrel down off target. The second is that Bersas trigger is a bit longer and has more of a forward sweep than most other guns. These things combined throw most people off. You will need to adjust your trigger pull to the gun.
Now that you know the what and why. It will be up to you to work out the correction at the range. The good thing is. Once you get it figured out you will be able to shoot your Bersa and other Bersa guns without issue. I can go from my Bersa Thunder HC Pro 9mm to any of my other guns and not have to consciously adjust. Once you learn what to do you just automatically do it when you have the Bersa in hand.
I stumbled on the problem after several trips to the range with my 380. I had more than one gun with me that day and laid them all out and looked at the triggers, which is when I discovered the forward hinged issue. It is not a huge difference but it is enough that it throws many people off.