Safari browser on iPad displays the list of recent searches when you start typing the letters on the search box. This is an useful when you want to pick some thing from the recent search list. But if you are using a shared iPad and you do not want to your searches to be in the recent list then you can use the clear option available as part of Bookmark.
Tap the Bookmark icon and then the Clear History button in History window.
This would display the following popover with Clear History option.
Tapping the Clear History button would clear the Recent Search list
Also Read: Date & Time and Time Zone settings on iPad