The case number doesn't help here, since this is mainly a user community forum, and nothing account-specific can be handled here.
How do you know they are going to put the credit on your credit card? It might be coming by check. Unless you gave the credit card info to TurboTax and that's what they told you they would do, they don't automatically have your credit card info unless you have the card info stored in your online TurboTax account. But even then it may come by check since your purchase was at Costco and not linked to your TurboTax account. A few years ago when I needed a Costco TurboTax refund for a duplicate purchase, TurboTax sent me a check in the mail. As I recall, it wasn't extraordinarily long--maybe 10-14 days. But that's anecdotal.
Did you use the online form to request your refund? If so, go back to the same interface and way down near the bottom is a tiny blue link "Track refund requests." I don't know if that states how it is coming or not.
