Don’t have a new page button on my iPhone 10?

The document Turn Private Browsing on or off on your iPhone or iPod touch - Apple Support has not been updated yet for iOS 15

For iOS 15, see the following from Browse privately in Safari on iPhone - Apple Support

(and if there is no new page button at the bottom right of your screen when you are in a website, scroll up for it to appear)

You can use Private Browsing Mode to open private tabs, which don’t appear in History on iPhone or in the list of tabs on your other devices.

1. Tap .
2. Tap  in the bottom center of the Tab bar at the bottom of the screen, then tap Private.
3. The tab is automatically added to a Tab Group called Private. You can open multiple private tabs in the group.

You can easily confirm that you’re in Private Browsing Mode by checking that the search field bar is gray or displays the word Private.

To hide the sites and exit Private Browsing Mode, tap , then tap  to open a different Tab Group from the menu at the bottom of your screen. The private sites reappear the next time you use Private Browsing Mode.

To close private tabs, tap , then swipe left each of the tabs you want to close.

An easy way to create a new page is by swiping to left on the url bar on the bottom of the screen. Or you can grab the url bar and swipe up like entering app switcher on an non homebutton iphone