privacy protip

I don't know about your DNS provider farming queries through your search engine (I guess by fetching the content of the visible results?) but I do know that if you have search suggestions on then everything you type is sent to your search engine and recorded before you even hit enter.

Supposedly DuckDuckGo doesn't retain this data but I've been out of the loop so that may have changed at some point. I've heard conflicting stories about their privacy and I haven't dug into it yet.

For DNS providers though you can check out these: https://www.privacyguides.org/en/dns/#recommended-providers

I'm not entirely sure of the situation with firefox. At the very least, if you are searching for a single word then firefox will check for existing TLD domains that use that word. Those lookups can be seen in DNS logs. If the TLD exists then the browser autocompletes the domain name and the browser redirects to the autocompleted url. If the TLD doesn't exist you get redirected to the corresponding default search provider search results, but the DNS logs still have your TLD lookups.

With chrome and chrome-likes, it doesn't really matter because you've already decided to give up your info. For all we know, Google could be logging your search query directly out of the search box before sending the text to your preferred search engine. It's Google - if you can imagine it, they're probably already doing it.