The Germans had a huge problem with building either turbosuperchargers or turbojet engines. They lacked suitable quantities of nickel alloys that were required. While nickel is one of the most common metals in the Earth's crust, mineable deposits are fairly rare. The only one that came close to being under German control was in very northern Finland, and hard to get to under the best of circumstances. Even today, nickel is a relatively rare and valuable metal; in the 1990's the Russians even considered converting one of their Typhoon subs into an ore carrier to get nickel from arctic sources by going under the icepack.
In order to build jet engines in quantity with inadequate alloys the Germans had to sacrifice sevice life. The typical Me-262 or Ar-234 jet engine had a life of around 25 hours, and keeping track of how many hours an engine had was vital - so much so that when the Americans captured German jets after the war they often found the Germans had destroyed the engine records so to keep us from figuring out which engines were usable. Thus, even under non-combat conditions, flying captured German jets was like riding a time bomb. In combat it was even worse.
The Germans admired American turbochargers but did have the alloys to build them. They tried with the FW-190 and found you needed the alloys for not only the turbochargers but also the exhaust pipes leading to them. They tried substituting steel pipes with an aluminum coating (a method still used today) with less than stellar results.
This problem carried over to other weapons as well. The fabulous V-2 rocket engine used a cold gas reaction to drive the turbopump, using H2O2 in order to avoid the need for a high temperature gas generator. So did the Me-163 for its rocket engine.
And at first their need for high altitude performance was minimal. The FW-190 was noted for running out of power quite noticeably at around 25,000 ft and when Kurt Tank kept telling the Luftwaffe about his plans to fix that problem they replied, "Why should we care?" The RAF did not use much in the way of high altitude aircraft at that point (note that they had the turbosuperchargers yanked from the Lightning Mk.1). Then the USAAF 8th Air Force showed up in the summer of 1942 and it was a different story!
The Me-262 was never a real threat to the Allies because the Germans could never build enough of them or build them reliable enough to create a real threat. Hitler's famous "Build them as bombers" order never was a factor in the Me-262 attaining operational status, and probably was the correct use of the airplane in the summer of 1944, given its very limited numbers and limited service life. The USAAF alone shot down something like 175 Me-262's, far more kills than the German jets ever got.
As for info on US WWII turbos, I have some illustrated breakdown parts drawings.