No. Qualification for the stimulus is based on your 2020 tax situation.
The IRS used the 2019 tax return of tax filers in the past, merely to get payments out as quickly as possible. But that does not change the fact that one's qualification for the stimulus payment is based on your 2020 tax status. For example, if you qualify to be claimed as a dependent on your 2020 tax return (which you will not file until 2021) then you do not qualify for the stimulus.
If you were an undergraduate college student in 2020 then chances are you DO NOT qualify for the stimulus payment and never will. It is most common for an undergraduate to qualify to be claimed as a dependent on their parent's tax return, and is extremely difficult for such a student to have sufficient ***EARNED*** income to support a claim they they provided more than half of their own support.
The IRS can look at your 2018 or 2019 return to issue a stimulus check. If they do not have a 2018 return then you may be waiting for them to finish processing 2019 before they will be able to use it to issue your stimulus check.
If you used the non-filer site, you will not be able to e-file a 2019 return after using the non-filer site.
TurboTax has no control over your stimulus check, nor can TT track it for you or tell you when the IRS will issue it. Refer to the sites linked below:
You can call the IRS hotline at 1-800-919-9835 to get updates on your stimulus payment.