JSFiddle - React, Tailwind, and code Playground

HTML

<input type='text' id='short'>

JavaScript

var $short = $("#short"),
    url;

$short.on('paste', function (e) {

    var pasteData = e.originalEvent.clipboardData.getData('text');
    url = pasteData;
    shorten();
});

function shorten() {
    $.ajax({
        url: '/your_script.php', // Pointed at your php script
        type: 'POST', // Or GET - totally optional
        data: 'url=' + url, // Your args
        dataType: 'text', // Define the type of data you will get back
        success: function (data) {
            // Data should be your echoed data
            // This is faking it
            $short.val(data);
        }
    });
}