By using event.preventDefault(), you're inhibiting the anchor tag from doing what it wants to do naturally, which is hyperlink you to another resource. As a result, you won't be redirected to your desired location.
That said, why not just add some additional Javascript to perform the redirection at the bottom of your click function?
http://www.tutorialspoint.com/javascript/javascript_page_redirect.htm
$(document).ready(function () {
$('.start-printing').click(function () {
event.preventDefault();
window.print();
var url = window.location.href;
var formId = url.split('/')[4];
var newPage = "";
if (url.split('/')[3] == 'form') {
newPage = 'form_2';
} else if (url.split('/')[3] == 'form') {
newPage = 'form_2';
} else if (url.split('/')[3] == 'form_2') {
newPage = 'form_3';
} else if (url.split('/')[3] == 'form_3') {
newPage = 'form_4';
} else if (url.split('/')[3] == 'form_4') {
newPage = 'form_5';
}
url = url + newPage + formId;
window.location = url;
});
});