When I check the status of my tax return on the TurboTax website, it shows as "approved," but on the IRS website, it is still not approved.

[DoninGA]

On the TurboTax Tax Home web page it would indicate Accepted, not Approved.  Accepted and Approved have two totally different meanings.

After the tax return has been Accepted by the IRS (meaning only that they received the return) it will be in the Processing mode until the tax refund has been Approved and then an Issue Date will be available on the IRS website.
See this IRS website for federal tax refund FAQ's - https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions