The Web Search is searching of the specific Individual that you have on your screen, along with dates and relationship AND the Star Setting that you have for the web search.
You may receive a lot of hits, but using the number of Stars may help reduce the number of hits. You can also limit the search in the type of records you are looking for in the upper right of the results screen.
The results my not always be in the same order. I rarely go beyond two or three pages of hits.
You can also use the Status Options, also in the upper right corner of the Results Screen.