There's an add-on for what you want to do
The "Google Container" add-on does wat you ask for:
How does it work?
Installing this extension deletes your Google cookies and logs you out of Google.
The next time you navigate to Google it will load in a new red colored browser tab (the “Container”).
You can log in and use Google normally when in the Google Container. If you click on a non-Google link or navigate to a non-Google website in the URL bar, these pages will load outside of the container.
Clicking Google Share buttons on other browser tabs will load them within the Google Container. You should know that using these buttons passes information to Google about the website that you shared from.
You can get it here
Even if you find some way to do what you describe, in the precise Google context you mention it wouldn't make any sense:
if you take a closer look at the Google search results (e.g. hover on the link and see the target displayed at the bottom of your browser window)**, all the clickable links lead to google.com, and have as parameters some session cookie plus the target website.
These are NOT direct links to your target website, it's Google's server which will redirect you to your destination !
So, even if you open those links in a private window, Google will still know its you that requested that link!
**EDIT: the mechanism is a little bit more subtle. At first, a hoover will display the link to your website. BUT as soon as you try to click them (try a right-click!) that link will be substituted with a link to google.com!
.. Google's engineers thought faster than you, and be sure that if today you find some hole in their tracking, tomorrow they will close it: it's their business, in ALL senses.
The only way to avoid search-engine tracking is to use a privacy- aware service such as duckduckgo.com or qwant.com