If the fed refund is $600 and you own $300 state, turbo tax sent in for you to be refunded $300 federal. Where is the balance of fed to pay the state $300?

Your state tax due cannot be paid from your federal refund.

It is not possible to pay your state tax due from your federal refund, nor is it possible to pay a federal tax due from a state refund.  The federal and state treasuries are completely separate from each other.  You may have seen a TurboTax summary of your federal and state amounts that confused you into thinking one was “paying” the other.  It is not.  If you have federal tax due you can pay by mailing your payment, having the payment taken out of a designated bank account, or you can pay directly on the IRS website.  You must pay your state tax due using the state’s preferred method of receiving payment.  For most states that will be by making a payment to the state’s own tax website, or by mailing a check or money order.

https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe