deleting search suggestions in site-specific searches

I have noticed that on certain websites which have site search boxes on their webpages (e.g. reddit, tvtropes), if I click in the box to type a search term, it will display about five suggestions which I have searched in the long-distant past in some context which I no longer recall. I generally would prefer suggestions not to show up in the first place, and I don't know why these specific terms are the specific ones which show up. Since this issue happens with site-specific search engines, rather than the search engine associated with the address bar, I'm unsure how to delete these suggestions. I normally browse in private windows, and on the rare occasion that I don't, I usually delete my history, and am unsure how the suggestions came to persist, and to persist only in the context I describe? If anyone can help, I'd be thankful.