There is a lot of old email search history stored on my Google and Internet Exploretr desktop. How do I delete it forever?

   There are a lot of histories.  I don't think any of them are called desktop history.  I would assume that Google keeps its history in the registry.  If you find where, you can delete it.
   Our program, RegVac, has a StashVac which may remove them.  If it doesn't, you can find where they are kept in the registry and have it remove them each time you run it.
   Internet Explorer has a few histories, so it depends on which one you want to remove.  From your description, my guess is that you want to remove the AutoComplete history (this is the popup which shows what you have previously typed when typing in a textbox).  That is not an easy thing to remove because it is kept under lock and key in the Registry by Windows.  About the only thing to do is to delete the item (select the item from the popup menu and then press the Delete button) when it pops up.  Our program, RegVac, has an AutoComplete section which looks at all of the AutoComplete entries and makes textboxes for each one, so that you can delete them.  You cannot just delete the whole section.
   Unless Google or Internet Explorer has options in their settings where you can disable the History (I don't think that the Histories that you are referring to, do), you cannot just delete them forever.  They always fill back up with what you type in their textboxes.