An increase in Altitude will result in a decrease of atmospheric pressure -1 in. Hg. / 1000ft (ISA conditions), but with a turbocharged airplane it will keep the pressure constant (+/- MSL pressure) up to the critical altitude? Why do we have to lean the mixture although pressure is kept constant or has it do with the fact that density decreases and thus reduce the amount of molecules for a given fuel molecule? What is the relationship between pressure and density? Thanks :-)
Turbo or not, you lean mostly because the mixture required for takeoff is far richer than necessary for lower power settings, even at sea level. Aircraft engines are partly air cooled and partly liquid cooled - the full-time liquid cooling is oil, and on takeoff, there is part time liquid cooling in the form of excess gasoline from the full rich mixture.
The reduction in air pressure with altitude, or density with heat, means this effect becomes more extreme the higher you go, but you should be leaning, even at sea level, once power is reduced from takeoff (you should be leaning on the ground as well as soon as the engine is running smoothly after start, until just before takeoff, if you want to best mitigate the effects of the lead in the avgas).
The turbocharger just fools the engine into thinking its altitude is lower than it really is. A cruise power setting of 22" of manifold pressure is 22" inches of manifold pressure to the engine, whether it's 22" of natural atmospheric pressure achieved at wide open throttle at 8000 ft on a non-turbo engine, or 22" of artificially produced manifold pressure at 15000 ft in the turbocharged one.
The density of the compressed air intake charge is reduced by compression heating of the charge by the turbo, which reduces its efficiency (you'll need a slightly wider throttle opening to get the same MP with a hot charge) and complicates cooling (the hot air reduces detonation safety margins). Intake intercoolers are used to reduce the temperature of the intake charge, without reducing the pressure, to mitigate this problem.
In all cases, leaning is done in reference to exhaust gas temperature, usually some value on the lean side of peak EGT to get best fuel economy and relatively cool cylinders, with an eye on cylinder head temperatures.
With a turbo, the main effect of altitude that has to be kept in mind is cylinder cooling efficiency. The turbo gives the engine an artificial ability to make low altitude power at high altitude. The problem is although the intake charge is "thick", being compressed air, the air coming in to cool the cylinders is "thin" and the heat rejection rate via the cooling fins declines, so leaning and cowl flap settings need to take this into account in managing cylinder head temperatures.
Exactly how that is to be done depends on the engine/airframe, per the POH and you should be leaning at all times based on POH procedures.
Cooling efficiency is critical to cylinder life with turbocharged engines, because they tend to operate a lot closer to temperature margins because of the demands of cooling in thin air. Good engine baffling is key. Badly installed baffles or degraded baffle seals, that wouldn't be a problem in a non-turbo installation, can cause high cylinder head temperatures with a turbo at higher altitude. Even if the cylinder head temperatures are good, bad baffles can lead to hot spots not picked up by the CHT probes and can ruin a cylinder with no obvious outward signs until it's too late.