Yes, if you see accepted, that means the return has been filed. Accepted means the government has received your return but not processed it yet. The next step would be for them to process and approve your refund. It can range from a span of just a few days up to about 3 weeks to see a move from accepted to approved. It will vary depending on how huge of a backlog they are working on when processing returns.
After you have filed successfully using TurboTax, the IRS and/or state handles all processing of the returns and your refunds. TurboTax no longer has any control nor does any processing/handling of the returns and refunds.
And you can use your State Department of Revenue's website to check status of
state tax refund. Choose your state here
How do I track my state refund? and it will give you steps to track the status.
If you claimed
Earned Income/Additional Child Tax credit on your return, the government held those returns until around mid-February. And refunds from those returns will start releasing in
late February. You can find more details on that here
Avoid the Rush Track Tax Refund Online at IRS.gov