The problem is that JS is broken in your site, most likely because you have JCH Optimize to bundle JS code. The comment pagination is normally using AJAX to reload the next page. In your case, when you click on the 2nd page, you see that the page is reloaded which means the JS code used for pagination is not triggered. So disable JCH Optimize's JS code optimizing/bundling and things should work as expected.

My bad, it's not related to JS errors as only the comment submission is powered by AJAX. You should actually see "?limitstart=" in comment pagination (you have "?start=") but even if you modify the URL manually, the page does not load the right comments.

Which makes me think that perhaps some plugin is messing up with start/limitstart. Can you check for any 3rd party plugins running in the K2 item view?