Hi
Error showing was set to "None" in joomla settings, but before I've checked that I've changed template to Protostar to test if the problem is coming from template. Unfortunately problem persists, so it's not a template based or because of some conflict with template.
During a look for a probable cause I had to move all unpublished modules to trash, so if You need any of them, that's where You'll be able to find it. Thanks to that I've discover that none of the modules alone is affecting the site.
Errors were shown by: "System - jbetolo", "System - Marco's SQL Injection - LFI Interceptor", "Authentication - JFBConnect Social Integration", "Content - JFBCContent", "System - JFBCSystem", "User - JFBConnect Social Integration", "Jomsocial Update", so I've disabled them in plugins list as well. So after hiding showing errors You need to enable all of them again.
Finally after couple of tests more I've disabled a module called "ExtraWatch Agent Module" and site started to work magically.