That is not how Private browsing works. From Apple's explanation:
You can use Private Browsing to open private tabs that don’t appear in History on iPhone or in the list of tabs on your other devices. Tabs in Private Browsing aren’t shared with your other Apple devices, even if you’re signed in to the same Apple Account.
Browse the web privately in Safari on iPhone - Apple Support
It doesn't mean that no data is saved to your device.
You can go to Settings>Apps>Safari>Clear History and Website data. But really, unless there's a problem, it's rarely worth the effort. The amount od data is generally very small.