In the control-settings, there is a button to "cycle" through your turrests. This button doesn't seem to function lin the way, the description says.... (in comparison: the broadsite-button speaks of "selelcting")

The "cycle" button will select the turret, but not cycle... (you select the turret-group. Projektile, beam or missile.) And if you manually control your turrets, you always fire the group at the same time. There is no way to control a single turret, if it is in a group.

Or am I just doing something wrong?

Yes, I found that out, too. I'm pretty sure, that different Lasers ARE grouped, however. Different missiles, arn't (just tried a swarm and shield-buster-turret. You can cycle through). And Flak and Pulse also not. (they both count as projectile. Although they have different icons. I don't know, if scatter and Pulse are grouped.) If you install 2 turrets of the same type (two pulse, for example), they ARE grouped, and function as one.
But I'll try the lasers again, perhaps I was wrong there, too.

Lasers ARE grouped. Single-Fire-Missiles seem to group, too. (at least a Muranth badger and shield-buster. The first is a rare weapon-type, you only can find as bounty.)

It's realy a shame, that this game isn't supported anymore. The different turrets should be grouped alone, imho. They can function VERY differently. Especially the lasers. And the Missiles proof, that this is possible.

the best way to do this is to put the same turrents that group on the same side of the ship, that way when you want to take control of lets say the left side of the ship, you can cycle and pick up that side, usually used when your trying to pound a captial ship. You'll be blasting with your broadsides, but you can also cycle and pick up all the grouped weapons on that side if they are capable of damaging a capital ship. I don't worry about fighters of course automated fire usually wipes them out easily.

This is good if you usually have your lighter weapons targeted for fighters only, once they are all dead you want those weapons, if they can also damage capital ships, to shoot the enemy big ships also. Many new players tend to forget about light weapons when they are trying to broadside fire on enemy big ships. Some lighter weapons can also shoot and damage them after the fighters are destroyed.

Plus, once in a while you'll stumble over big ships without any fighters about, so you want that small bit of extra firepower from lighter weapons.