All turbos are oil-cooled.
Some are water-cooled in addition though.
There's no problem with using an oil-cooled one, you just have to let it all cool down a lot longer after parking. When the engine is running there's enough oil flow through the turbo so the oil isn't in there long enough to be 'hurt' by the high temperatures. So you just let it cool down for a few minutes and they're fine.
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