[Malleys]

Так форма должна отправляться, если удалась валидация!

<form id="form_name" action="save_user.php">
	<input id="first_name" name="first_name" required>
	<input type="submit" id="btn_name">
	<output id="message"></output>
</form>

<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script>

	$(function () {
		$('#form_name').submit(function(e) {
			e.preventDefault();
			var first_name = $('#first_name').val();

			$(".error").remove();

			if (first_name.length < 1) {
				$('#first_name').after('<span class="error">This field is required</span>');
			} else {
				sendForm('form_name', 'form_pass');
			}
		});
	});

	function sendForm(form_name, form_pass) {
		var form = $("#" + form_name);
		$.ajax({
			url: form.attr("action"),
			type: "POST",
			dataType: "html",
			data: form.serialize(),
			success: function (response) {
				$('#message').html('OK');
				var result = $.parseJSON(response);
			},
			error: function (response) {
				$('#message').html('Error message');
			}
		});
	}

</script>