Online TurboTax saves each time you change screens.  Even if you were to close the browser screen or get disconnected your entries should be saved.

The best way to sign out is to go to the left menu column and scroll way down to the bottom of that column.  There is a  "Sign Out" link.

You may have to actually place the cursor in the left menu column to scroll down the menu column.  Scrolling the whole screen down may not be sufficient since the left column scrolls independently.