Safari (across iOS and OS X) remembers some of your recent searches and shows them to you when you start typing in the omnibar. It's aimed as a usability feature where you can quickly repeat your search.
The problem is this recent search history is visible to all. Anyone that logs into your Mac and uses Safari can see what you've been searching for. If that bothers you, you should clear recent searches from Safari. And it's possible.
Here is How to Clear Recent Searches from Safari in Mac OS X:
Step 1: Open a new tab/window in Safari
Step 2: Click in the URL bar.
Note:This should be done in order to view the “Recent Searches” menu. In case you are having any URL coming up, delete it. Otherwise you won’t be able to see the menu.
Step 3: Once you see the “Recent searches” list, scroll down and you should see “Clear Recent Searches.”
Step 4: Click on Clear Recent Searches.
At once, all the recent search terms should disappear from the list. If you happen to type a few letters in the omnibar now, you will get suggestions only from your bookmarks and history (depending on your configuration). The recent search terms will no longer be seen because they're deleted.
After a while though, the recent search terms show up. You will have to clear them out routinely if you want to keep them off the suggestion list.
Is there a way to permanently turn this feature off? Not that we know of other than the blanket-approach: Private Browsing. In Mavericks, Safari did not come with any feature that lets you prevent recent search history. And in Yosemite, the trend continues.