Private mode
Using iPad Pro and iOS 15. In safari there is no icon on the page of double square to access private mode.
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It has changed with iPadOS 15. See the following from Browse privately in Safari on iPad - Apple Support
Visit sites without making history
You can use Private Browsing Mode to open private tabs, which don’t appear in History on iPad or in the list of tabs on your other devices.
Tap 
, then tap Private.
While Private Browsing Mode is on, the Safari background is black instead of white and sites you visit don’t appear in History on iPad or in the list of tabs on your other devices. You can open multiple private tabs in the Private Tab Group.
To hide the sites and exit Private Browsing Mode, tap , then switch to a different Tab Group. The tabs reappear the next time you use Private Browsing Mode.