How do turrets level up in Fallout 4?

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.

They work randomly, but 90% of the time it depends on the level of technology you build them around. For example if you build them in a blank field they will most likely be level 1, less likely - level 3. If you build a fusion power generator leave the settlement then return and build turrets around the power generator they will be 90% level 7, less likely - level 5.

Build them then delete the power generator and you will have a bunch of level 7 turrets. I have noticed though that the random level 5 turrets pop up more if they are facing things in your settlement that can be broken or destroyed like plants or power generators, etc. Like the game is trying to save you from yourself by not having explosions aimed at them.

My personal experiences contradict the statements regarding distance and randomness being the factors involved, I had rank 5 and 7 turrets at "Sanctuary". I believe my "Gun Nut" and "Armorer" perks may have affected my turret placements, but also I was a high level, 49.

It seems silly to potentially randomly generate rank 5 turrets at low level or depending on settlement area.

Also to note, I had a Super Mutant raid at Sanctuary, with high level Super Mutants using Missile Launchers.

It seems to me, that the higher your defence level for the settlement, the higher the number. Once I hit #7, I could switch them all out to #7. I honest don't think they're any more powerful.

It is true that the closer you are to Sanctuary, the lower the number on the Heavy Machine Gun Turrets, and the higher the number the more powerful they are, mk7 has explosive rounds, the mk5 has incendiary rounds, still not sure about the mk3.

However, one thing I have learned is once you get a mk7 to spawn in a multi #ed settlement, if you stand right next to an originally spawned mk7, you can spawn the mk7s, two at a time (rinse and repeat) all day long.

This method probably also works with the mk3 and the mk5 but I haven't tried it.