How to delete messages off an Iphone 4s?

Okay so I have an Iphone 4s. I know how to delete messages that are in your inbox, that is easy. But i went to the far left screen (the search bar screen) and typed one letter and all my deleted messages came up and I could read them all there although they had been deleted. Is there a way to make it so you can't see my deleted messages on the search bar?

It never happens to me. This is the tip. Hope that help.