Why is google recaptcha box appearing in the top of the fields, It should be appearing between the inputs and the Confirm button. Take a look at this SS right here: https://i.stack.imgur.com/93G69.png .
<?php
$public_key="";
$private_key="";
$url="https://www.google.com/recaptcha/api/siteverify";
if(array_key_exists('submit_form',$_POST))
{
$response_key=$_POST['g-recaptcha-response'];
$response=file_get_contents($url.'?secret='.$private_key.'&response'.$response_key.'&remoteip='.$_SERVER['REMOTE_ADDR']);
$response=json_decode($response);
}
?>
<!DOCTYPE html>
<html>
<head>
<title>Authentification</title>
</head>
<body>
<form method="post" action="auth.php">
<center>
<fieldset style="width:300px;">
<legend align='center'><h1>Connexion</h1></legend>
<table>
<tr>
<td><b>Email: </b></td>
<td><input type="email" name="email" required=""></td>
</tr>
<tr>
<td><b>Mot de passe: </b></td>
<td><input type="password" name="password" required=""></td>
<tr>
<div class="g-recaptcha" data-sitekey="<?php print $public_key; ?>"></div>
<script src="https://www.google.com/recaptcha/api.js"></script>
<td><input type="submit" name="submit_form" value="Se Connecter"></td>
<td><a href="ajout.php"><button type="button">S'inscrire</button></a></td>
</tr>
</table>
</fieldset>
</center>
</form>
</body>
</html>