Can You Still Make A U-Turn Where A "No Left Turn" Sign Is Posted?

If a sign that says a you turn if prohibited, then no.  You have to make a left to MAKE that you turn. The same goes for a right turn you turn.  If the cops catch you making a you turn where you are NOT allowed to, you will get a fine & trust me, it ain't going to be cheap.

Not if it involves making a left turn.

If you are in the USA, and in the left lane, your you-turn would require a left turn and then some.  No, it would not be legal to make that turn if "no left turn" is posted.