The TurboTax software license only allows you to install the software on up to 5 computers that you own. You would need to sign onto your TurboTax account for each new installation.
If using the same computer to start another tax return all you need to do is click on the File menu in the upper left of the desktop program screen and click on New Tax Return. There is no need for you to sign onto your TurboTax account for another tax return using the same computer.
