After the user submits a form, it is usually a good idea to display a confirmation page. It might be jarring to go directly to the privacy page. I would set it up like this: index.php?page=contact -> shows the form index.php?page=contact-submit -> shows the confirmation with privacy policy text
The privacy page is just an example. Your scenario replaces it with a confirmation page. So the problem still stands: How do you go to the ?page=contact-submit page? Do you use header()? Is that the canonical way?