Question

What I am trying to do is
On “Page 1”, user fills a form, and when he clicks on “Submit”,
a new page, say “Page 2” should get automatically created (in the same folder as index.php) in such a way that name of the page is say first_name.php (what the user filled on page 1) and it should display the contents of what he filled in on the first page.
Something like showed in the figure.

Answer 1

showName.php:

<?php

   echo "    Your name is {$_REQUEST['name']}";


?>

This is under the assumption that your form contains an input with name=name

So a form like this:

<form action="showName.php">
   Enter your name: <input name="name" />
</form>

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Eventually you might want some validation on the submit, but for now this should suit for what you want to do.

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Update:

I read your question more carefully —

You should NEVER do what you proposed it could lead all kinds of attacks and hacks on your site.

You could use a mod-rewrite to seem as if the page went to john.php but really it is showing showName.php?name=john