Magento 1.9 – Resolve jQuery Conflict Issue

Magento uses Prototype JS which has already claimed the $ variable. You need to use jQuery in "no conflict" mode so jQuery won't also claim the $ variable:

<script>
jQuery.noConflict();
</script>

After this, you cannot use $() to use jQuery, because then you call Prototype. You have to use the full name: jQuery(). You can change your self written scripts by:

1. Replacing all uses of $ by jQuery

2. Wrap your scripts in an anonymous function passing jQuery as a parameter:

   (function($) {
        // You can simply use $ here
        $('selector').dosomething();
   })(jQuery);
   

The jquery.lazyload.js library is well written and is able to operate in both normal mode and conflict mode. There's no need to change the files of this library (replacing all uses of $ by jQuery) as it is already making use of the anonymous function method, see jquery.lazyload.js:

16.  (function($, window, document, undefined) {
17.      var $window = $(window);
18.  
19.      $.fn.lazyload = function(options) {

             // *SNIP* (lots of code...)

240.     });
241. 
242. })(jQuery, window, document);

You can add local copy of jquery after following line[skin/frontend/yourpackage/yourtheme/js/jquery.min.js]

<action method="addJs"><script>mage/translate.js</script></action>
<action method="addJs"><script>mage/cookies.js</script></action>
<action method="addItem"><type>skin_js</type><name>js/jquery.min.js</name><params/></action>

OR

Use their js using following way:

<action method="addJs"><script>mage/translate.js</script></action>
<action method="addJs"><script>mage/cookies.js</script></action>
<action method="addItem"><type>external_js</type><name>http://www.owlcarousel.owlgraphic.com/assets/vendors/jquery.min.js</name><params/></action>

And rest of your code