I have completed/filed 2019 tax forms. Can I start with TurboTax (for 2020) by entering the data from these forms?

Each tax year is a separate TurboTax program.

Tax year 2020 will not complete until after 12/31/2020.  The TurboTax desktop CD/Download software for 2020 should be available in November 2020.  The TurboTax online editions for tax year 2020 should be available in December 2020.

What you need is a bookkeeping system  ...

 QuickBooks Self Employment bundle you can check out which includes one Turbo Tax Self Employed return and will help you keep up in your bookkeeping all year along with calculating the estimated payments needed ....
http://quickbooks.intuit.com/self-employed