It highly likely head gasket the issue, check some basics such as water pump and thermostat housing/hoses for leaks and also radiator and turbo cooling lines (visible coloured staining marks left from antifreeze)
You can use chemical test kits detect combustion gases in cooling system (we use gas analyser over rad filler neck)
It done some miles and to be honest I wouldn't want do just gaskets at that mileage, bearing shells, all new seals, new oil pump , new rings/pistons and valve overhual tends be best way forward but you could indeed do just head gasket and basic head overhaul if the short block seems decent enough and assemble with new water pump and quality japan timing kit (ICP do these at sensible money)
Good parts and good effort won't be that cheap & at this mileage further issues could crop up from fatigued engine stressed from the poor cooling/overheating .