AFAIK this always needs an STC.
Nobody has ever done one for a TB20. I vaguely recall reading that one of the US firms quoted something like $200k for doing an STC. Some owner(s) tried to put a group together but failed.
My favourite aftermarket turbocharger STC is for the Jacobs engine Cessna 195 – not quite a radial fire breather as the supercharged Super Stearman, but apparently the 195 will show 200 knots at FL200 with a 350hp turbocharged shakey Jake.
There are some 195s with P&W, I am guessing power application on take off is mellow to avoid shooting for the weeds.
It can be done.
I have a Commander with an aftermarket turbocharger. It isn’t rocket science, but paperwork is a killer.