I can't get my iPad out of the private browsing mode

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.

See the following from Browse privately in Safari on iPad - Apple Support

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.

You are very welcome.

The iPad User Guide may be helpful in getting to grips with the new features of iPadOS 15. It’s a rich source of often overlooked information and guidance. The Guide is available both online, using a web browser (such as Safari), or as an Apple Books download:

iPad User Guide - Apple Support

https://books.apple.com/book/id1567104892

In the image that I’ve attached above, the button is 303 Tabs (highlighted in blue) as this illustration has 303 open tabs - “n” representing 303.

You own browser is likely to have far few open tabs, so the number (represented by “n”) will be different.