Is there a way to keep the blue JIRA bar at the top (where you can switch between JIRA / Confluence, etc.) - when using the Application Navigator to visit a website that is not associated to Atlassian?
Unfortunately there isn't. The Application Navigator is essentially just a collection of links (internal or external). Since the JIRA top bar is associated directly with the Atlassian webpage. My suggestion would be just to open your external links in a new tab.
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