>This is an intended behaviour if you are using the same Google Account for your Gmail and Chrome. If yes, you'll be signed out of Chrome when you signed out of Gmail account.
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Good news is that it is possible currently to disable the feature. Doing to breaks the link between the Google Account in Chrome that is used to sync data and Google accounts on Internet sites.
Note: Google may remove experimental flags like the one described below at any time. As long as it turns up when you run the steps below it is supported.
Here is what you need to do:
1. Load chrome://flags/#account-consistency in the browser's address bar. Google Chrome should display the flag Identity consistency between browser and cookie jar at the top.
2. Set the flag to disabled with a click on the menu and selecting disabled from the context menu.
3. Restart the Chrome browser.
Chrome breaks the link between the Google account in Chrome used to sync data and Google accounts that you sign in using the browser on Google sites."
https://www.ghacks.net/2018/09/08/disable-the-sign-out-link-...