Are you using the download software? You do not "log in" and "log out" of desktop. Go up to File on the top of your computer screen and save your return, and next time you want to work on the return, launch the program from your own hard drive. Then you might have to go to File and click "open return".
The icon that you described is in the desktop TurboTax software, not TurboTax Online.
Follow the instructions that xmasbaby0 gave you to save your tax return. Click Save on the File menu, not the File tab. Or if you are using TurboTax in Windows you can press Ctrl-S. When you exit from the return or close the program you will be prompted to save.
With the desktop software, your tax return is saved in a file on your computer. TurboTax will provide a suggested file name, but you can use any name you want. You are responsible for keeping a backup of your tax return data file on removable storage or in the cloud in case the original is lost or damaged, or your hard drive or computer fails. TurboTax does not have a copy of your data file.