Form validation evaluation to avoid exploits
I think it looks ok - though someone might want to add on if they notice any issues I may have missed. However, a bit of cleanup goes a long way: 

# Set a 500 (internal server error) response code.
function outputResponse($override = "", $code = 500){
    # http_response_code($code);
    echo $override ?: "Oops! Something went wrong and we couldn't send your message.";
    exit;
}

# Exit early while doing negative checks
if(empty($_SERVER['HTTP_X_REQUESTED_WITH'] || 
    strtolower($_SERVER['HTTP_X_REQUESTED_WITH']) != 'xmlhttprequest' || 
    $_SERVER["REQUEST_METHOD"] != "POST"){
    outputResponse();
}

# 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);
$message = strip_tags(trim($_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.
    outputResponse("Oops! There was a problem with your submission. Please complete the form and try again.", 400);
}

# Set the recipient email address.
$recipient = "email_here";

# Set the email subject.
$subject = "New contact from $name";

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

# Build the email headers.
$email_headers = "MIME-Version: 1.0\r\n";
$email_headers .= "Content-type: text/html; charset=utf-8\r\n"; 
$email_headers .= "From: $name &lt;$email&gt;\r\n";
$email_headers .= "Reply-To: $email\r\n";
$email_headers .= "Return-Path: $email\r\n";
$email_headers .= "Organization: Bilingual Counselling\r\n"; 

# Send the email.
if (mail($recipient, $subject, $email_content, $email_headers)) {
    # Set a 200 (okay) response code.
    outputResponse("Thank you! Your message has beent sent.", 200)
} else {
    outputResponse();
}