Url problem due to htaccess
This is not really a drupal question. You are seen the website through a frame 
The real url is http://31.186.174.215/~leenkneg/ where the urls work as expected.

To answer your question after the info you gave.
One solution is to add an A record to the DNS of the domain. So every http traffic will land to your site but without the users notice and the can have the emails where they want.

So you will have the hosting and the can keep the emails.