When I exit private mode in Safari, all of my tabs go away. How do I take all of my tabs with me out of private mode?
As of iOS 12 beta 2 and below, this is not currently possible with Safari.
As said by @Snowbody,
[You can copy & paste] all the URLs from the URL bar into an email or other document[.] Of course that doesn't preserve history, form fills, etc.
Your best option is to either find a different browser with that feature (maybe create a new question) or to submit a feature request to Apple.