How do you implement reCaptcha in php/html?

I'm having this problem where reCaptcha is not validating and not running my php code. I have one php file, and one html.
The code is supposed to send an email if the reCaptcha is validated, but ends up doing nothing and freezes on a blank screen on my webpage when i hit submit.

html cut-out:

<form onsubmit="" action="../structure/process_email.php" method="POST">


    



    



    





    

</form>

reCaptcha script is placed in header:

https://www.google.com/recaptcha/api.js

php cut-out:

<?php 
$headers = "MIME-Version: 1.0\r\n";
$headers .= "From: xxx <xxx@xx.xx.xx>\r\n";

    if(isset($_POST['submit'])) {
        require_once('../structure/recaptchalib.php');
        $privatekey = "xxxxxxxxxxxxxx";
        $resp = recaptcha_check_answer ($privatekey,
        $_SERVER["REMOTE_ADDR"],
        $_POST["recaptcha_challenge_field"],
        $_POST["recaptcha_response_field"]);
                            
        if ($resp->is_valid) {
        	$email_to = 'xxx@xx.xx.xx';
		$email_subject = 'New message: BMS - Contact us';
		$name = $_POST['name'];
		$email = $_POST['email'];
		$message = $_POST['message'];

		$content = "From:"." ".$name."\n"."Email:"." ".$email."\n\n".$message;
		mail($email_to, $email_subject, $content, $headers);
		header('location:../index.html');
        } else {
        }
   	header('location:../contact_us/contact_us.html');
    }
?>

I have tried many different approaches on this problem, and i can't seem to find any solution. Have i forgot to write some code somewhere?