How do tow trucks transport mangled messes?

Flat bed trailers.

You have never seen a deck truck wrecker?
They winch the mangled wreck up on the deck which tilts then lash it down and off they go.
For bigger wrecks they winch it up on deck trailers.

@all Thanks .

They extract the mangled bodies first.

Not all the time @Yellowdog I have known several wrecker drivers that have transported the wreck back to the shop with the dead body still in it, and extract the body out of it there.
Of course the coroner is on hand at the extraction.
Some cases it takes to long on a bust highway.

My uncle drove a tow truck in St Louis several years back.
He told a gruesome story about having to help rescue workers remove a long steel pipe which went through the driver and the back of their seat. The other end was out the windshield, or where it was before being shattered.