Difficulties with displaying succes message after form is sent

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Javascript

HTML

PHP

Ajax

I'm still learning the code languages that I used to create my form with. So I can't seem to figure out what seems to be the problem. Here's my code:

HTML:

<form action="" method="POST">
    <ul class="form-style-1">
        <li>
            <input type="text" id="mail-name" name="name" class="field-divided" maxlength="15"  placeholder="Voornaam *" /> <input type="text" id="mail-lastname" name="lastname" class="field-divided" maxlength="15" placeholder="Achternaam" >
        </li>
        <li>
            <input type="email" id="mail-email" name="email" placeholder="E-mail *" class="field-long" maxlength="40" >
        </li>
        <li>
            <input type ="text" id="mail-phone" name="phone" placeholder="Telefoonnummer" class="field-long" maxlength = "15">
        </li>
        <li>
            <select name="subject" id="mail-subject" class="field-select" >
            <option disabled value="" selected hidden >--Onderwerp-- *</option>
            <option value="Kennismakingsgesprek">Kennismakingsgesprek</option>
            <option value="Meer informatie">Meer informatie</option>
            <option value="activiteit">Aanmelding activiteit</option>
            <option value="Vraag/klacht">Vraag/klacht</option>
            <option value="Contact">Overig</option>
            </select>
        </li>
        <li>
            <textarea name="information" id="mail-information"  placeholder =" Je bericht *"class="field-long field-textarea" maxlength="2000"></textarea>
        </li>
        <button class="mail-submit" id="mail-submit" type="submit" name="submit">Send e-mail</button>
        <span class="form-message"></span>
    </ul>
</form>

JS:

<script>
   $("#mail-submit").click(function(event){
    event.preventDefault();
    var name = $("#mail-name").val();
    var lastname = $("#mail-lastname").val();
    var email = $("#mail-email").val();
    var phone = $("#mail-phone").val();
    var subject = $("#mail-subject").val();
    var information = $("#mail-information").val();
    $.post("contacttest.php",
        {
          name: name,
          lastname: lastname,
          email: email,
          phone: phone,
          subject: subject,
          information: information,
          submit: "yes" 
        },
        function(data){
            $(".form-message").html( data );
        }
    );
});   

PHP:

<?php

if (isset($_POST['submit'])) {


    $email_to = "#";

    $email_subject = "#";

    $name = $_POST['name'];
    $lastname = $_POST['lastname'];
    $email = $_POST['email'];
    $phone = $_POST['phone'];
    $subject = $_POST['subject'];
    $information = $_POST['information'];


    $errorEmpty = false;
    $errorEmail = false;

    if (empty($name) || empty($email) || empty($subject) || empty($information)) {
        echo "<span class='form-error'>Voer alle velden in!</span>";
        $errorEmpty = true;
    }
    elseif (!filter_var($email, FILTER_VALIDATE_EMAIL)) {
        echo "<span class='form-error'>Geef een geldig E-mail!</span>";
        $errorEmail = true; 
    }
    else {
        $formcontent=" Naam: $name \n\n Achternaam: $lastname \n\n Email: $email \n\n Telefoon: $phone \n\n Onderwerp: $subject \n\n Informatie: $information";
        $mailheader = "From: ".$_POST["email"]."\r\n"; 
        $headers = "From: ". htmlspecialchars($_POST['name']) ." <" . $_POST['email'] . ">\r\n";
        $headers .= "Reply-To: " . $_POST['email'] . "\r\n";
        $headers .= "MIME-Version: 1.0\r\n";
        $headers .= "Content-Type: text/html; charset=ISO-8859-1\r\n";
        mail($email_to, $subject, $formcontent, $mailheader);
        echo "<span class='form-success'>E-mail has been sent!</span>";
    }

}
else {
    echo "Not working!";
}

?>

What I'm looking for is that the page doesn't refresh after the form has been sent, so that it can display the form-succes or error message. At the moment, the page doesn't refresh(I think), but it refuses to show the succes message when the form is sent. All of the code are located in 1 PHP file.

Thank you for your time.

What I have tried:

Basically looked through the internet about this problem, didn't fix my problem.