failure when clicking on thumbnail

Could you try to switch your template to the default joomla template and see if you get the same result ?

I see that your template uses the javascript library jquery. There are often issues between jquery and mootools (the javascript library in joomla). So it could be related.

Unfortunately, when that happens on your website there is no js error thrown and I couldn't find any clues on the web

Well, the solution would be to find why that happens.
Because it's not only there. It's also on the checkout on the address selection screen where the use can edit/create addresses in a popup.
Basically, anywhere where a popup is used, it doesn't work on your front end because of apparently a conflict between both libraries.

What to do ?
1. Ask for help on the mootools / jquery forums. They might know what you could do to fix the problem.
2. Remove the javascript jquery files from your template (physically as well as the code loading them in the index.php). You don't seem to be using anything from it currently. It is used in your template to display some effects on sub menus but you don't have submenus.