There are no e-filing fees using the TurboTax online editions. The TurboTax account fees are for the use of the software to assist in completing a tax return. The account fees are the same whether a tax return is e-filed or printed for mailing.
Only the TurboTax online Free edition is free to use and is limited to very simple tax returns that do not require any other forms or schedules other than the Form 1040 - https://turbotax.intuit.com/personal-taxes/online/
The TurboTax online editions for tax year 2020 should be available in early December.
Turbotax Online is "free to start." The final price depends on the complexity of your return. Simple returns will have free preparation and e-filing for federal and state taxes.
If your income is less than $36,000, you can use the IRS "free file" version which is free even if you have complicated forms.
If you have more complicated forms, you will need to pay for the preparation service (use of the program). There is no extra fee for e-filing after you have prepared your return, but there may be a preparation fee.
Turbotax Online is not currently turned on for 2020, expect it to be available in mid-December.
To install Turbotax on your own computer, you will have to buy a copy.