Turbochargers can be damaged by dirty or poor oil. Most manufacturers recommend more frequent oil changes for turbocharged engines. The turbocharger will heat when running. Many recommend letting the engine idle for several minutes before shutting the engine off.
This gives the turbo time to cool down and will increase the life of the turbo.
The time required for the turbo to build up the needed pressure is called turbo lag. This is noticed as a hesitation in engine response. This is caused by the time taken for the exhaust system to speed up the turbine. The directly-driven compressor in a supercharger does not have from this problem. Lag can be reduced by using lighter parts. This allows the turbine to start faster. Other mechanical changes can reduce turbo lag, but at an increase in cost.