Frankly, your example doesn't sound like typical search behavior. Searching the text "sale" in my org brings me all layers with "sale" anywhere in their title, descriptions, or tags.
You can also reference the REST docs to see other ways of more specifically searching for things. For example, searching "title:sale" will only return items which have the string "sale" somewhere in the title. This is equivalent to a "contains" or "ILIKE '%sale%'" query elsewhere.
Do you consistently see your search only do a "begins with" query? That seems like it shouldn't be happening.
The lack of wildcard support is a bit of a shame, though. But if you wanted to do a thorough search of your org's content, the Python API can provide you a bit more functionality.
- Josh Carlson
Kendall County GIS