I think you're confused about the terminology for the TurboTax products, and it's confusing the people who are trying to answer your question. If you use TurboTax Online you access it in a web browser. You do not download or install anything.
The TurboTax software that you install on your own computer is called the CD/download software or the desktop software. You can buy it on a CD or as a download. You install it on your computer from the CD or download. You run the software on your computer. You do not use it online.
TurboTax Online and the CD/download desktop software are completely separate products.
TurboTax Online, that you use in a web browser, is only available for the current year. TurboTax Online cannot open a .tax20yy file that was created by the CD/download desktop software, and it cannot open tax returns from previous years. You cannot download the TurboTax Online software. If you need to open a .tax20yy file you need the CD/download desktop software for the corresponding year.
If you buy a download of the desktop software from TurboTax.com the download will remain available in your account for 4 years, so you can download it again if necessary. You could also just save the download along with your tax return files on your computer. The download is just one file.
