I've tried leading, trailing, target only, fire at will, and every combination of them, the targeting modes don't affect their accuracy.
While the "catching up with ships" is exactly what I said they do because of the fatigue mechanic.
It's the fact they need to "catch up" in the first place that's the problem.
I got curious enough to check it myself again and here is my experience.
I took my Corvette and installed 5 turrets on it:
x2 M Beams with Efficient upgrade and Thermal shock experimental (thermal heat)
x1 L Beam laser with Long range upgrade and Thermal shock experimental
x1 S pulse laser with Rapid fire upgrade and Scramble Spectrum experimental (malfunctioning)
x1 S Multicanon with High capacity upgrade and Corrosive munitions experimental (extra 25% damage on hull and 20AP to all weapons)
I switch targeting mode to trailing and mostly was flying with fire at will. x1 L Beam laser is the slowest turret on a ship and it has the worst ToT of all turrets which is understandable as L slot on Corvette is absolute trash and no gimbal nor turrets works well there. It's a slot for fixed weapons, torpedoes (if someone using them) or missiles. However, because it was long range beam, it has a lot more ToT than most turrets due to range. So if you want to put any turret in L slot on Corvette, from my experience long range is the best bet because it's the only way to increase it's overall ToT. L slot doesn't work well in close combat is what I am saying.
Still, it's an optimal set of turrets for WEP capacitor with 0-2 pips
The "fatigue" mechanic from my experience only depends on enemies abilities and your ship positioning. My turrets were accurate enough for my taste. I would say it was at worst 75-80% ToT and 85-90% ToT at best. Which is a good result in my opinion. Best turrets are definitely combination of 2M and 2S at all times. L beam laser was very effective because of long range. S multicannon was MvP when it comes to hull and it was only used when shields are down. But I think L slot is still more suited for fixed weapons. Or maybe it's a slot for some heavy turret like Beam OC for example or for Multicannon C3 with Corrosive.
In my test these turrets were effective against all kinds of targets. But usually I've been dealing with highly experienced and engineered pirates and terrorists from mission board.
Now, the real problem is that turrets can't keep up with ships on boost. It is probably mechanically balanced, so turrets are lagging behind ships that are boosting. However I have fixed that because in my case enemies were not able to boost as I was constantly heating them with Thermal Shock and they had all kinds of malfunctions as well. Still, I would say that turrets are not there for damage, but as utility weapons. Which is why I am using Thermal Shock and Scramble Spectrum as a measure to suppress enemies ability to use SCBs, to suppress boosting and ability to return fire while Corrosive munitions are there to maximize output of all turrets when dealing with Hull. Yes, enemies under severe heat stress won't be able to use SCBs effectively, won't be able to boost, won't be able to return fire with weapons like railguns or plasma. Thermal shock (if you have enough of it) helps you to suppress returning fire and other abilities of ships in combat.
Does works in PvP as well, but needs a lot more intensive heat sources and very different approach to build. After all, heat can be used against you when facing Plasma Conduit builds. However, ships like Python MK2 and FDL which are relying on Plasma Conduit the most, can't withstand all that heat damage for long periods of time, so if they can't kill you in that time frame even with boosted damage from Conduit, they would be cooked eventually. Literally.
There might be better combinations of turrets, I also have x2M pulses with rapid fire and thermal shock which might be better than M lasers. Those turrets might be even more effective at heating than Beams, but needs testing.