Sending Email with Ajax, its not sending but saying Thank You! Your message has been sent

Question

Sending Email with Ajax, its not sending but saying Thank You! Your message has been sent, but not getting any email, i tried on gmail and hotmail both are not working.I have posted the form and mail.php and main.js, these are the only files for the email. I do not seem to know what the problem is, Its executing correctly but not sending email.

index.php:

 <form class="contact__form" method="post" action="mail.php">
                                
                                    <!-- form message -->
                                    <div class="row">
                                        <div class="col-12">
                                            <div class="alert alert-success contact__msg" style="display: none" role="alert">
                                                Your message was sent successfully.
                                            </div>
                                        </div>
                                    </div>
                                    <!-- end message -->
    
                                    <!-- form element -->
                                    <div class="row">
                                        <div class="col-md-6 form-group">
                                            <input name="name" type="text" class="form-control" placeholder="Name" required>
                                        </div>
                                        <div class="col-md-6 form-group">
                                            <input name="email" type="email" class="form-control" placeholder="Email" required>
                                        </div>
                                        <div class="col-md-6 form-group">
                                            <input name="phone" type="text" class="form-control" placeholder="Phone" required>
                                        </div>
                                        <div class="col-md-6 form-group">
                                            <input name="subject" type="text" class="form-control" placeholder="Subject" required>
                                        </div>
                                        <div class="col-12 form-group">
                                            <textarea name="message" class="form-control" rows="3" placeholder="Message" required></textarea>
                                        </div>
                                        <div class="col-12">
                                            <input name="submit" type="submit" class="btn btn-success" value="Send Message">
                                        </div>
                                    </div>
                                    <!-- end form element -->
                                </form>

main.js:

(function ($) {
    'use strict';

    var form = $('.contact__form'),
        message = $('.contact__msg'),
        form_data;

    // Success function
    function done_func(response) {
        message.fadeIn().removeClass('alert-danger').addClass('alert-success');
        message.text(response);
        setTimeout(function () {
            message.fadeOut();
        }, 2000);
        form.find('input:not([type="submit"]), textarea').val('');
    }

    // fail function
    function fail_func(data) {
        message.fadeIn().removeClass('alert-success').addClass('alert-success');
        message.text(data.responseText);
        setTimeout(function () {
            message.fadeOut();
        }, 2000);
    }
    
    form.submit(function (e) {
        e.preventDefault();
        form_data = $(this).serialize();
        $.ajax({
            type: 'POST',
            url: form.attr('action'),
            data: form_data
        })
        .done(done_func)
        .fail(fail_func);
    });
    
})(jQuery);

mail.php:

<?php

    if ($_SERVER["REQUEST_METHOD"] == "POST") {

        # FIX: Replace this email with recipient email
        $mail_to = "haroutdeurdulian@gmail.com";
        
        # Sender Data
        $subject = trim($_POST["subject"]);
        $name = str_replace(array("\r","\n"),array(" "," ") , strip_tags(trim($_POST["name"])));
        $email = filter_var(trim($_POST["email"]), FILTER_SANITIZE_EMAIL);
        $phone = trim($_POST["phone"]);
        $message = trim($_POST["message"]);
        
        if ( empty($name) OR !filter_var($email, FILTER_VALIDATE_EMAIL) OR empty($phone) OR empty($subject) OR empty($message)) {
            # Set a 400 (bad request) response code and exit.
            http_response_code(400);
            echo "Please complete the form and try again.";
            exit;
        }
        
        # Mail Content
        $content = "Name: $name\n";
        $content .= "Email: $email\n\n";
        $content .= "Phone: $phone\n";
        $content .= "Message:\n$message\n";

        # email headers.
        $headers = "From: $name <$email>";

        # Send the email.
        $success = mail($mail_to, $subject, $content, $headers);
        if ($success) {
            # Set a 200 (okay) response code.
            http_response_code(200);
            echo "Thank You! Your message has been sent.";
        } else {
            # Set a 500 (internal server error) response code.
            http_response_code(500);
            echo "Oops! Something went wrong, we couldn't send your message.";
        }

    } else {
        # Not a POST request, set a 403 (forbidden) response code.
        http_response_code(403);
        echo "There was a problem with your submission, please try again.";
    }

?>

Thank you in advance!