How do turrets level up in Fallout 4?

Doing some research on the matter, I find the reports to be conflicting; However, it appears that turrets do not level up. Instead, they spawn at a particular level, randomly. Furthermore, it appears you can have Mk 1, Mk 3, Mk 5 and Mk 7 turrets.

Discussion and research in this Reddit thread suggests that each settlement has a "maximum level", in which the turrets will spawn. It is reported that this maximum level increases, the further you are from Vault 111. This is most likely the case, with an emphasis on how far you are to the south-east. The farther south-east you are, the harder the encountered enemies are. The harder the enemies are, the tougher your turrets need to be, to help push them back.

What I can confirm, which appears to be the best course of action in aiming for the best defense, is that you can still rig which turrets you build. As detailed in this Steam Guide, entitled "Get The Best Turret For Your Settlement", the turrets level generates randomly when you select it from the workshop menu.

As such, you can move between the Machinegun Turret and Heavy Machinegun Turret, and each time, the selected guns level will randomly generate. By rotating the turret to the left, you can see what number is written on its wing, before you build it. If it is not the highest desired level, simply move back and forth between your defense options, until the desired level spawns.

Others have basically answered the question, here I am gonna provide some explanations for the mechanism.

In Fallout 4 (same for Fo3, FNV, Skyrim, etc.), the map is divided into different "Encounter Zones", and for each zone there are parameters controlling the minimum and maximum levels of NPCs spawn there. This means the turret level you can get depends on the zone that your settlement is in.

Nonetheless, if you are willing to use console command (PC only), then you can spawn turrets of desired level in any settlement. The command to be used is PlaceLeveledActorAtMe (detailed explaintion here). To do the trick, you need an ActorBaseID of the turret:

Machinegun Turret         B3036 
Heavy Machinegun Turret   1167A8 

and an EncounterZoneID whose level threshold is high enough (Here is a complete list of the EncounterZoneID), 1FA57 (Spectacle Island) suffices since you can build a Mk7 turret there normally. Now for example, command

player.PlaceLeveledActorAtMe 1167A8 3 1FA57

will spawn a Heavy Machinegun Turret MK 7 right next to your character, you can then use the building menu (the turret has to be in the settlement area, of course) to move it to any desired place.

The only decent thread I could find on the subject links the turret version to the average level of the enemies in that area. Basically, the turrets scale with the area you are in. So in the settlements that are far away from Vault 111, you will get the higher versions of the turrets, while the closer settlements like Sanctuary will usually only have the Mk I versions.

The subject is discussed at the bottom of this Reddit thread.