Your code is overwriting the
websites
property of the items in your source array with a filtered list.
Change it to return the filtered list instead:
const results = data?.map(singleData => singleData.websites.filter(website => website.name.toUpperCase().includes(searchQuery.toUpperCase())));
NB: You should generally normalize strings to upper-case for comparison, to avoid
the "Turkish i" problem[
^].