Safari Private Browsing doesn't work with Google search engine

Private Browsing is private from other folks with direct local access to your device. Not you from sites you access.

For privacy, consider enabling iCloud+ Private Relay, and maybe switch to DuckDuckGo for searches.

You appear to have been misled. "Private browsing" does not prevent Google from tracking you.

Browse privately in Safari on Mac - Apple Support

You are not alone in harboring that misconception:

"65% of these respondents reported feeling “Surprised”, “Misled,” “Confused,” or “Vulnerable” upon learning about the limitations of Private Browsing."

Source: https://spreadprivacy.com/is-private-browsing-really-private/

Don't use Google.