Thank you for using Apple Support Communities. It's my understanding that you'd like to know about the small cloud icon that appears when using Private Browsing on your iPhone. I'm happy to help.
The icon you're seeing means you're using iCloud Tabs to access open tabs in Safari on other devices using the same Apple ID. Please review the information and links below:
Browse the web - iPhone User Guide
View tabs that are open on your other devices. If you sign in to iCloud with the same Apple ID on all your devices, and you turn on Safari in Settings > [your name] > iCloud on iPhone, you can view tabs that are open on your other devices. Tap , then scroll to the list at the bottom. To close a tab, swipe left, then tap Close.
Set up and use iCloud Tabs
Have a great day!