TurboTax has no control over how long the government takes to process returns. They do not send any further updates to TurboTax after they accept your return.
The next step is for them to review/process your return once it's in their possession. And once they process the return and approve your refund, they will update the status with a deposit (or check mailed) date for you, on their website.
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.