What browser are you testing with? I loaded your home page, and the Fan module is showing properly for me after I wait quite a few minutes. I've tried reloading multiple times.
I do see another inclusion of the Facebook Javascript library on your page. That may be causing the issues you're running into. The following block of code should be disabled from whatever extension is adding it:
<div id='fb-root'></div><script type='text/javascript'>
// Load the SDK Asynchronously
(function(d){
var js, id = 'facebook-jssdk'; if (d.getElementById(id)) {return;}
js = d.createElement('script'); js.id = id; js.async = true;
js.src = '//connect.facebook.net/tr_TR/all.js';
d.getElementsByTagName('head')[0].appendChild(js);
}(document));
JFBConnect is loading a similar, but different, block of code directly underneath that.
Finally, you shouldn't need jQuery Easy just for JFBConnect. It shouldn't cause problems though. If you have problems with jQuery conflicts and JFBConnect, enable the "Include jQuery / Bootstrap" setting in the JFBConnect -> Configuration area. That will load a compatibility mode version of those libraries specific to JFBConnect that shouldn't conflict with anything else.
Thanks,
Alex