.php page loading when sending email from contact form (emails sending fine)

Question

I'm not very good at php (yet) and my contact form somehow seems not to work properly when I hit send: my .php page appears with the message that should appear on top of the contact form itself (in the div called "form-letyouknow"). I have made contact forms using this code before with different ids so I am sure it's a very simple and stupid error but I've been over it so many times now that I need someone with fresh eyes to look.

Here is my code:

        <div id="form-letyouknow">
      <div id="form-messages"></div>


      <form class="form" id="contact-form" method="post" action="send.php">


        <div class="margin-bottom">
            <label id="namelab" for="name">Nom:</br></label>
            <input type="text" id="name" name="name" size="40" required>
        </div>

        <div class="margin-bottom">
            <label for="email">Adresse e-mail:<br></label>
            <input type="email" id="mail" name="email" size="40" required>
        </div>

        <div class="button" id="submit">
            <label for="message" id="message-label">Message:<br></label>
            <textarea id="msg" name="message" rows="8" cols="50" required></textarea>
        </div>

        <div class="field">
            <button type="submit" id="send">Envoyer</button>
        </div>
      </form>
    </div>

Jquery

    $(function() {

// Get the form.
var form = $('#contact-form');

// Get the messages div.
var formMessages = $('#form-letyouknow');

// Set up an event listener for the contact form.
$(form).submit(function(e) {
    // Stop the browser from submitting the form.
    e.preventDefault();

    // Serialize the form data.
    var formData = $(form).serialize();

    // Submit the form using AJAX.
    $.ajax({
        type: 'POST',
        url: $(form).attr('action'),
        data: formData
    })
    .done(function(response) {
        // Make sure that the formMessages div has the 'success' class.
        $(formMessages).removeClass('error');
        $(formMessages).addClass('success');

        // Set the message text.
        $(formMessages).text(response);

        // Clear the form.
        $('#name').val('');
        $('#email').val('');
        $('#message').val('');
    })
    .fail(function(data) {
        // Make sure that the formMessages div has the 'error' class.
        $(formMessages).removeClass('success');
        $(formMessages).addClass('error');

        // Set the message text.
        if (data.responseText !== '') {
            $(formMessages).text(data.responseText);
        } else {
            $(formMessages).text('Oops! An error occured and your message could not be sent.');
        }
    });

});

PHP

<?php
    // POST requests.
if ($_SERVER["REQUEST_METHOD"] == "POST") {
// Get the form fields and remove whitespace.
$name = strip_tags(trim($_POST["name"]));
$name = str_replace(array("\r","\n"),array(" "," "),$name);
$email = filter_var(trim($_POST["email"]), FILTER_SANITIZE_EMAIL);
$message = trim($_POST["message"]);

// Check that data was sent to the mailer.
if ( empty($name) OR empty($message) OR !filter_var($email, FILTER_VALIDATE_EMAIL)) {
    // Set a 400 (bad request) response code and exit.
    http_response_code(400);
    echo "Oops! Il y a eu un problème. Avez-vous bien remplit tous les champs ?";
    exit;
}

// Set the recipient email address.
// FIXME: Update this to your desired email address.
$recipient = "[email protected]";

// Set the email subject.
$subject = "New message via website from $name";

// Build the email content.
$email_content = "Name: $name\n";
$email_content .= "Email: $email\n\n";
$email_content .= "Message:\n$message\n";


// Send the email.
if (mail($recipient, $subject, $email_content)) {
    // Set a 200 (okay) response code.
    http_response_code(200);
    echo "Merci ! Votre message a été envoyé. J'y répondrai dans les plus brefs délais.";
} else {
    // Set a 500 (internal server error) response code.
    http_response_code(500);
    echo "Oops! Une erreur est survenue. Vous pouvez m'envoyer un e-mail à l'adresse suivante: [email protected]";
}

} else {
// Not a POST request, set a 403 (forbidden) response code.
http_response_code(403);
echo "Une erreur est survenue lors de l'envoi de votre message. Réessayez ou envoyez moi un e-mail à l'adresse suivante: [email protected]";
}


?>

Thank you for taking the time and sorry if it's a dumb mistake.