JSFiddle - React, Tailwind, and code Playground

HTML

<div>Before sort: <span id="before"></span></div>
<div>After sort: <span id="after"></span></div>

JavaScript

var MyArray = ["13 Jun", "09 Jun", "25 Aug", "30 Jun", "13 Aug"];
document.getElementById("before").innerHTML = String(MyArray);

(function () {
    var ad = new Date(),
        bd = new Date(),
        months = {
            Jan: 0, Feb: 1, Mar: 2, Apr: 3, May: 4, Jun: 5,
            Jul: 6, Aug: 7, Sep: 8, Oct: 9, Nov:10, Dec:12
        };

    MyArray.sort(function (a,b) {
        var as = a.split(' '),
            bs = b.split(' ');

        ad.setDate(as[0]);
        ad.setMonth(months[as[1]]);
        bd.setDate(bs[0]);
        bd.setMonth(months[bs[1]]);

        return ad - bd;
    });
})();

document.getElementById("after").innerHTML = String(MyArray);