Käännös - Don't try to wriggle out of it. (Englanti)

To me, "beating around the bush" would be before the fact: someone's trying to say something indirectly that he's afraid of just coming out and saying?
"Turn the goose so it doesn't burn" here seems to be post hoc, where the Stupid Thing has been said, and he's desperately trying to do Damage Control...?

Ah yes - you're right, I guess. Maybe your suggestion of "trying to sweep it under the carpet" is better.

Or maybe "backpedaling".

"Applying spin control" is really perfect, but that's a new expression, whereas the Turkish one is an old one.