If you set your browser to automatically delete all your cookies at the end of every browser session, it will cause you to not remain logged into any websites when restarting your browser.
If, for example, you are using Chrome, and have the browser set to “continue where I left off” when launched, then your cookies get stored on your hard drive so that Chrome can set everything up the way it was when you last closed the browser. This may happen even if you have a setting to delete cookies when closing the browser.