Yes, you can track the progress of your refund using the Where's my refund? tool.
If you want additional information beyond that, you can all the IRS. TurboTax doesn't get any additional information about your refund after it is accepted.
Note, the IRS didn't start processing e-files this year until January 29, so filing two and a half weeks before that date doesn't do anything to speed up the process.
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