Implement AJAX ‘load more’ with offset posts

Your query doesn’t know anything about which posts you’ve already displayed. All it knows is that the second page of a 4 posts per page query is logically going to be posts 5-8. If you want to get a specific range of posts you should use offset instead of paged:

$args = [
    'offset'.        => absint( $_POST['offset'] ),
    'post_status'    => 'publish'
    'post_type'      => 'post',
    'posts_per_page' => 4,
];

$ajax = new WP_Query($args);

You’ll just need to keep track of the appropriate offset on the front end.