Deleting items saved in search bar -Yahoo, Google

When I begin typing a subject into the search bar (not
the address bar)there is a list of all the words I have
used in searching.

I cannot find any easy way to delete these. I feel it is
another way that someone else using my computer could
have some idea of my search interests.

I do know that Internet Options, clear history etc. will
get rid of saved sites but it doesn't solve my stated
request.