Hi
Ok. I found the problem. Apparently in some cases an update does not remove/replace old files which means You have like part of the new 3,7 and parts of older version trying to work together.
In an attempt to fix this I have manually overwritten joomla from a clean installation. Unfortunately even that haven't fixed the problem.
On joomla.org forums I have found couple of tips how to fix it:
https://forum.joomla.org/viewtopic.php?t=938117 or
https://forum.joomla.org/viewtopic.php? ... f567aa783dBoth those threads mention using
https://myjoomla.com/is/secure scanner to narrow the problem. Unfortunately I've already used my free trial.
Anyway problem is strictly connected with joomla and not our products, so I'm afraid this is as far as I could help. Try talk to joomla developers.
PS: I have found some malware scripts inside Your installation. Removed couple of them, but it would be best to do the scan to eliminate it all.