URL Rewrite or at least URL Redirect

Hello

I want to rewrite the domain part of an URL to redirect the trafic from one site to another.

To be more specific, for privacy reasons i will use an self hosted instance of SearxNG to anomyse all my search queries.

But because i dont trust my users to not use google, i want to rewirte www.google.com to www.mysearxngdomain.com (not a real site, just an URL Exemple).
The reason that i want a Rewrite and not a redirect its because i want to keep the page portion of the URL so an search like httpw://www.google.com/test becomes https://www.mysearxngdomain.com/test 

I dont know it its possible.But if it is will be great. If not, a URL redirect will be better than nothing.

I used workaroud using the notification page of an Website category to use <meta http-equiv="Refresh" content="0; url=">www.mysearxngdomain.com" />  as notify page and blocked google on the web policy. This works like an redirect but it's very clumbersome and not seamless at all.

Can you guys help? Can i make the rewrite or at least the redirect?

Thanks