I've been having this strange issue. I transferred my site to a different hostname and now when I open site's pages while being logged in Admin panel of Joomla, it logs me out right away. But only in the same browser. Also I've noticed that when I'm using that particular site, each opened page adds a new line in db's "session" table (under user0) Other sites just add one single line (user0) and that's it (in addition to admin's name when I'm also logged in into Joomla backend simultaneously) Other sites that reside side by side (virtual hosts) are fine on the same physical computer within the same OS (which is Ubuntu 16.04 in my case).
This sounds like a cookie domain issue. Open the configuration.php file located at the root of your site and clear the values for
public $cookie_domain = ''; public $cookie_path = '';
Next, close your browser to end the existing sessions. Now try again.
Note: configuration.php normally has read-only file permission. So change the file permission before making the changes.