Replace Iframe with Ajax

There is no way to achieve this, due cross domain security restrictions in modern browsers. You aren't allowed to access other domains content via ajax.

The other point is, that you may have css bugs if you load an entry page in a div on your site, because the loaded css will override your site's css.

Assuming the websites you are 'previewing' are on the same domain as your website, you can amend every a tag within the loaded div to perform an AJAX request with the following code:

var $container = $('#websitePreview')
$container.load('http://www.example.com'); // initial load

// link click within loaded content.
$container.on('click', 'a', function(e) {
    e.preventDefault();
    $container.load($(this).attr('href'));
});

On the other hand, if the previewed sites are on separate domains, you cannot load HTML from them due to the Same Origin Policy. An iframe or CORS request is your only option in those situations, and if the url is 3rd party, the latter is extremely unlikely to happen.

try

$('div#websitePreview a').live('click', function(event){
   event.preventDefault();
   var url = $(this).attr('href');
   $('div#websitePreview').load(url);
}