This isn't caused by a virus or malware. You don't need to install anti-virus on your iPhone.
What you're trying to get rid of is Google's search box search history. The search box remembers what you've searched on this specific phone if you're not logged into the Google website with a Google account, or on any of your devices if you
are logged in. I'm assuming you're not logged in, as if you were you'd use your Google account settings page to delete your search history.
You
should be able to delete individual past searches from the drop-down box by using the X button, but it looks like there's some kind of a bug in there that's currently preventing it (I tried it myself just now and ran into the same problem). It might be an incompatibility between the Google website and the iOS WebKit, which is used by all web browsers on iOS.
Go here:
https://www.google.com/history/optout
Turn off Signed-out search activity
It
should forget everything you searched for on the Google website in the past, on the phone you're using. It did on mine anyway.
If you want Google to remember your searches again, turn the setting back on.