It may seem unimportant to you but "private browsing" is not "private searching".
Even in a private window or tab your search terms are sent out to the world, and depending on your chosen Search Engine are used to assemble a digital image of "you" that is retained, sold, etc just the same.
Change your default search engine on iPhone - Apple Support
The scam web page "safenetgroup.site" is not a consequence or result of using Private Browsing. It's just one example of pathetic scams that can arise on any device anywhere at any time: Phony "tech support" / "ransomware" popups and web pages - Apple Community.
Clicking the "Run a test" button won't actually do anything, other than redirect to the App Store in an attempt to get you to buy some garbage, perhaps preceded by some equally pathetic animation to make it appear it is doing something. Go ahead and tap it for entertainment's sake if that's your idea of a good time.
Read Browse the web privately in Safari on iPhone - Apple Support. If whatever you seek to accomplish by using Private Browsing is not explicitly described in that document, then you may be laboring under a misconception.