Are you using mouse only? On my controller I usually use both analog sticks to move and turn the screen at the same time which allows quick rotation if necessary. Same applies imho to mouse and keyboard. I use the mouse to turn the screen and wasd to turn my character which at least for me leads to the same result.

A lot of comments missing this part: you have to lift your mouse to relocate it back to center. A low sensitivity mouse results in so called "rowing".
There are a lot of different preferred styles, as an example: eSports tends to higher DPI (dots per ich) since every millisecond counts but the precision falls behind which can be trained.

After all I would recommend trying a mainstream shooter game like Counter-Strike and leave the settings vanilla.
Start a match or training or what else and pick yourself 3-4 points of the environment like a corner of a wall, a bench and so on.
Now aim at one point, close your eyes and try to move your cross hair onto another point, open your eyes and you will notice that you are either too far beyond the aimed point or didn't reach it. This offset can be adjusted inside the most in-game setting by decreasing or increasing the mouse sensitivity.
If you always have to crank up or down your sensitivity in every game to the same direction you probably should change your mouse DPI. Most mice come with software where you can adjust this. Also when playing on windows disable mouse acceleration.