Hello Kenneth Koppervik,

Please ignore my previous reply.
After we further debug your site, it seems like you have enabled "'--enable-magic-quotes" in which this will effect the ajax to work.
Can you please disable the magic quotes and check your site is the issues still persist?
Please do update us then,

If you have anymore question, please don't hesitate to ask us.
Please be advise.
Thanks.

I've got this set up on several other cPanel accounts - all with magic quotes enabled. They work. So that's not it.

I had to set up the site from the bottom up again and it's working now. I couldn't find out what was causing the conflict.

Hello Kenneth Koppervik,

I am really sorry for the delay of this reply as it is a weekend for us here.
Is it possible temporary disable "'--enable-magic-quotes" from your server and see the Media manager is work properly in your Joomla backend or not?
Because we need to rid out of which possibility causing this issues, so we can continue to investigate on this issues
Hope you understand.