Normally, the ‘Private’ button appears in the lower left corner of the Safari browser when a new tab is opened (Read Turn Private Browsing On/Off in Safari). However, there are a couple things that can cause private browsing button not to be there.
Missing ‘Private’ Button – Restricted Websites
The ‘Private’ button will not show up on the browser if the restricted websites option is turned on.
1. Settings > General > Restrictions
2. You will be prompted to enter your passcode. Press ‘Websites’
3. Press ‘All Websites’
The private browsing mode will now be restored. Open a new tab to see the ‘Private’ button.
Missing ‘Private’ Button – No Favorites
There ‘Private’ button will also not show up if there are no links in the Favorites. You will need to add something to your Favorites for the ‘Private’ button to appear. (Read Manage Favorites for Safari)