Why can’t I see the refund owed for my state return?

You are mixing terms. A refund is what you get back from the state if you over withheld taxes and had more withheld than you needed to so the state sends the overpayment amount back to you as a refund.

If you owe, that means you didn't have enough state taxes withheld from your pay and you must now make up the underpayment by paying the state what you still owe. 

Are you looking for whether you are getting a refund or whether you owe? You should be able to see those number on your state tax return. It would say something like refund or overpayment for the refund. It would say something like taxes owed or taxes due for what you still owe.

The amount should also be at the top of the screen in TurboTax when you are completing your tax return for the state.