Allow form to send POST request but prevent page from reloading - Javascript

If you include the query strings in the script url:

<script src="https://www.google.com/recaptcha/api.js?onload=onloadCallback&render=explicit"async defer></script>

then you can use grecaptcha.getResponse as it says in the google reCAPTCHA documentation:
https://developers.google.com/recaptcha/docs/display

<script src="https://www.google.com/recaptcha/api.js?onload=onloadCallback&render=explicit"async defer></script>
<script type="text/javascript">
  var verifyCallBack = function(response) {
    alert(response);
  };
  var widgetId;
  var onloadCallback = function() {
    widgetId = grecaptcha.render('recaptcha', {
      'sitekey' : 's',
      'theme' : 'light'
    });
  }
</script>

<form>
  <div id="recaptcha"></div>
  <input type="submit" name="submit" value="submit">
</form>

$('form').submit(function(e) {
  e.preventDefault();
  var response = grecaptcha.getResponse(widgetId);
  $.ajax({
    url: 'validate_captcha.php',
    type: 'POST',
    data: {'g-recaptcha-response': response},
    success: function(data) {
      alert(data);
    },
    error: function(error) {
      alert(error);
    }
  });
});

And then in validate_captcha.php:

<?php
if (isset($_POST['g-recaptcha-response'])) {
  function CheckCaptcha($userResponse) {
    $fields_string = '';
    $fields = array(
        'secret' => 'secret_key',
        'response' => $userResponse
    );
    foreach($fields as $key=>$value)
    $fields_string .= $key . '=' . $value . '&';
    $fields_string = rtrim($fields_string, '&');

    $ch = curl_init();
    curl_setopt($ch, CURLOPT_URL, 'https://www.google.com/recaptcha/api/siteverify');
    curl_setopt($ch, CURLOPT_POST, count($fields));
    curl_setopt($ch, CURLOPT_POSTFIELDS, $fields_string);
    curl_setopt($ch, CURLOPT_RETURNTRANSFER, True);

    $res = curl_exec($ch);
    curl_close($ch);

    return json_decode($res, true);
  }
  $result = CheckCaptcha($_POST['g-recaptcha-response']);

  if ($result['success']) {
      echo 'success';

  } else {
     echo 'error';
  }
}
?>

So now in your javascript, you can use the data variable inside success:function(data) in an if statement:

if(data == 'success') {
  registerUser(name, email, password); // not a real function, just an example
}