No, Turbo Tax can not check if it was approved. Only Accepted for processing. After it is Accepted they are done with their part. The IRS doesn't update Turbo Tax or anyone else about the status of your return. You just have to wait or ask the IRS.
If your return was Accepted or mailed then check IRS Where's my refund
https://www.irs.gov/refunds/
When will you get your refund?
https://ttlc.intuit.com/community/refund-status/help/when-will-i-get-my-irs-tax-refund/01/25589
IRS - Questions about Refunds
https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions
