How to remove private browsing mode from iPad

See the following from https://support.apple.com/guide/ipad/browse-the-web-privately-ipad8ea0fc1a/ipados

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.

Note that doesn't completely remove the possibility of using private browsing again -- it only exits private browsing mode. Also note that "Start Page" qualifies as a "different Tab Group" in the above instructions.

The method of accessing Private Browsing differs between iPadOS13/14 and iPadOS15.

These instructions relate to iPadOS13/14 only - as the support page has not been updated for iPadOS15:

Turn Private Browsing on or off on your iPad - Apple Support

The method used to access Private Browsing has changed for iPadOS15. To access Private Browsing in iPadOS15, it is necessary to first summon the Safari Sidebar using the button at upper left of the screen [1] - and from the menu tap the Private toggle button [2].

To return to regular browsing, tap the sidebar n Tabs button, that appears above the Private button.