Is there a way to search privately on Google without Chrome?
 

or using another browser.

What I mean, is there a page like private.google.com or incognito.google.com that I can use that won't save search history.

He asking for private search on google, not for other search engine lol.
So here is the solution:
https://www.startpage.com/

Uses google search results, but without privacy violations.

Ha. I actually already use Startpage however, I don't think that they are using exactly the same search alogorithm.

I use Startpage as the default page within my browser and it's fine for what I use it for however, I am starting some new stuff and I am going to be heavy searching and I don't think SP will cut it.
So, I just need incognito Google but without having to set google as my default search engine.

Thought they would have one. A bit like the way Wikipedia has a simple version simple.wikipedia.org

Try this one as well: https://www.dogpile.com

I haven't used dogpile since about 2002 and then only once but it's actually pretty decent. Isn't it?

"Startpage is the only one whose search results come from Google, due to a unique and longstanding agreement in which Startpage pays the search giant to get a feed of links for any search. The result is a search engine that feels a lot like Google did before it leaned into personalized search and advertising."

Is it possible the differences you are seeing are from personalized search?

Google also shows different result per country, possibly even per region if big country. So if startpage shows results for one country only that could be reason as well.

The difference for me is more than just site ranking.
I know Google loves Youtube but Startpage is something else.
I can literally see a full page of nothing but Youtube with many searches.

Just delete your search history after its performed so your wife and law enforcement can't find it.

And what about all the linked content on devices logged in with the same ID?