Old wives tales. Modern turbo engine are very good and very durable IF properly maintained… that same can be said of ANY engine. None need be “cooled down” before shutting off the engine since all are oil and water cooled now. The turbocharged gasoline auto engine has been around for 60 years (surprised?) so they’ve had a LOT of time to work out the kinks.
I’ve owned 3 built in 3 different decades from 3 different manufacturers from 3 different countries.
1985 4 cylinder turbo bought new, drove 83,000 miles, no problems, sold the vehicle. Lots of lag but a blast to drive.
2001 4 cylinder turbo bought as a demo with 4K miles drove it 104K more miles NO turbo problems at all. Almost no lag, BIG broad power curve very well suited to its automatic transmission, got 32 mpg highway with a 5 passenger sedan.
Current 5 passenger 4 cylinder turbo bought used with 33K now with 40K. Best engine yet. Gets 34 highway, best acceleration of the 3 and the smallest engine of the 3 at 2.0 liters.
From someone whose owned 3 over 35 years, turbo engines can be as reliable as any normally aspirated engine. It all depends on the manufacturer and the maintenance.