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modern automotive turbo chargers typically are water jacketed - they are cooled by the coolant system. the old turbochargers from the 1980's and older lacked this, and were only cooled by the oil. one must let the car idle a minute or two for the turbo cool down. otherwise, the trapped oil in the bearing would congeal, and eventually plug the passageway, and the turbo would seize. the water jacket keeps the temperatures in check. synthetic motor oils can be heated about 200 degrees F higher (than conventional oils) before they start congealing, so this and the water jacket have eliminated the idling cool down. you may recall turbocharged cars from that era needing a replacement turbocharger after 35-70k miles, because few people adhered to the cool down time. if the car was allowed to idle down, the turbo would outlast the life of the engine.
naturally the aircraft engines would be lacking in providing a water jacket cooling system, with the exception of the DA-42 and the liquid cooled continentals.
turbos should also be given a proper warm up time. when the oil is cold, there really is inadequate oil flow to the bearing. this isn't a factor so much in aircraft engines as the taxi out and run-up gets a least a little heat in the oil. this is more a factor for the cars, and throttle applications should be minimal to keep the turbo from spinning up as much as possible for the first few minutes.
Not all modern turbocharged cars are cooled this way. The 3 of my cars which are turbocharged are all oil cooled. However, you bring up a good point about synthetic oils - we can get away with this in cars by using synthetic oils. There are no synthetic oils available to purchase for airplane engines. I don't buy the argument about 1930s technology not needing advanced oils - this here is a perfect argument in favor of synthetics. My guess as to why there aren't any is cost. Too much cost to get it approved and not enough market to recoup the money spent for design and approval.
Bottom line is that the powerplant vendors and manufacturers are going to have to do something sooner rather than later if avgas's days are numbered.