I know it's purpose is to prevent interstage leakage of pressurised gases, but then shouldn't the compressor stages have it as well? Why is it exclusive to the turbine stages?
The shroud fulfills two purposes: It improves aerodynamic efficiency at the blade tip (reduced tip leakage, due to better sealing possibilities with fins etc.), and it also changes the blade's vibration eigen-modes, because the shrouds interlock. The single blade dynamics changes to blade dynamics of the whole row, the blades get coupled.
Some more elaboration on why some turbine blades have shrouds and others don't, i.e. issues with shrouds:
Centrifugal load: the weight of the shroud needs to be carried by the blade, under significant centrifugal load, and at high metal temperatures (often close to some creep limit design criterion)
Blade dynamics: adding weight to the tip of the blade lowers the eigenfreqency significantly, and also changes other dynamic mode shapes - and you need to stay outside excitation frequencies.
Cooling: depending on the turbine stage, the shroud will need cooling as well, so there is a trade-off whether the increase in stage efficiency is enough to justify the increased cooling air consumption, and how much is left in the end
Why compressor blades usually don't have shrouds:
Blade dynamcis: Compressor blades are usually already build as thin as possible (for aerodynamic efficiency), while maintaining a minimum thickness for mechanical stability and vibration dynamics. Adding a shroud would be quite a challenge to blade dynamics.
Tip leakage: in compressors, the thermal deformation between hot and cold state is not as large in compressors as in turbines, which means that the tip gap can be designed tighter. Secondly, in compressors the pressure ratio per stage is a lot lower, so the tip leakage isn't driven by as much pressure ratio. That said, to my knowledge, the tip leakage is one of the main limiting factors modern compressors, both for efficiency and compressor stability / surge margin