You most likely have 'Restrictions' enabled on websites. Go to Settings > General > Restrictions and check if it says 'On'. If it does, enter you PIN and go to the 'Allowed Content' section. Under this section there is an entry for 'Websites'. Tap this and select the 'All Websites' option.
This should bring back the Clear History button as well as the Private browsing button.
This is actually quite clever on part of Apple. This is a form of parental control where you enable restrictions on the websites that can be visited and then obviously you would want to check on the actual websites that have been visited. Hence, Apple disables the Clear History and Private browsing options. Apple obviously has hidden this fact, deep. There could be a better way of doing this but, this is what we have for the time being.
Hope this helps!