The processing times vary for each return. If you e-filed the Virginia return, it's generally about 4 weeks for processing those. But if you mailed it, that takes around 8 weeks. Sometimes the processing can be delayed for a few reasons. And it also depends on if they have a backlog they are working on.
If you go here https://www.tax.virginia.gov/wheres-my-refund and click What can slow down your refund?, it will show you reasons that could cause a delay.
But they should provide you a status update on their Where's My Refund page too. Once they accept your return from TurboTax, they do not send any further updates on your return or refund to TurboTax.
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