Private Browsing

You can't delete it. Check Safari/Preferences/General to make sure Safari opens with is not set to Private Window,

you may be able to do this from Parental controls in OS X.

Change the account to a parental

and select the radio button for "Try to limit access to adult websites" and see if the option from Private browsing is enacted.
I played with this a little last night but gave up because there is too much overhead and simply blacklisting sites from the hosts table was far easier and less "parental" looking to the users I support.

Please open safari preferences , click on general set safari opens from new private window to new window .

it is permanent method to browse in normal window .

You can also use private browsing still normal browsing window is opened by pressing shift + command + N

And if in safari preferences , new private window is enabled , you can close the window press + w , then again click on safari from dock - normal safari browsing window opens .

Thanks !