How are they sending links - Well-formed anchor tags or just URL in regular text?

Try this:

Code:
<script>
       function sendFunction() {
           var userName = document.getElementById("myUserName").value;
           var userinput = document.getElementById("myInput").value;
           var imagesrc = document.getElementById("baseimage");
 var img = $('<img width=" =100px;" height=" 100px">').attr('src', imagesrc.value);
        
    $('#conDivID').append("<span style='color: red'>" + userName + ': </span>');
    $('#conDivID').append("<span style='color: blue'>" + embedUrls(userinput) + ' </span>');
    $('#conDivID').append('</br>');
    $('#conDivID').append("<span class='timeSpan'>" + timestring + '</span>');
               $('#conDivID').append('</br>');
               $('#conDivID').append(img);
               $('#conDivID').append('</br>');
               

              
           }

    function embedUrls(text) {
        var urlRe = /(https?:\/\/[^\s]+)/g;
        return text.replace(urlRe, '<a href="$1">$1</a>');
    }          
</script>