If you do Shift-Space to pilot with the mouse, the main weapon will automatically aim towards where the target is, including their movement. You'll see a little dot on screen showing where it's aiming. It's still not perfect, especially if you're strafing, and in travel mode too, but the moment your enemy maintain trajectory and speed for a moment, you'll hit. Keep in mind that different weapons have different projectile speeds, and will also have a certain level of inaccuracy.

Also, larger ships do have ACTUAL turrets.