Display Posts by Views (Popular Posts Widget)

It could be fix using session, try following.

Initialize session using following code in your functions.php.

    function init_sessions() {
    if (!session_id()) {
        session_start();
    }
}
add_action('init', 'init_sessions');

add a function for setting session

function set_session_posts_viewed($postID)
{
    if(empty($_SESSION['posts_viewed']))
    {
        $posts_viewed = array($postID);
        $_SESSION['posts_viewed'] = $posts_viewed;
    }
    else
    {
        $posts_viewed = $_SESSION['posts_viewed'];
        $posts_viewed[] = $postID;
        $_SESSION['posts_viewed'] = $posts_viewed;
    }
}

change post count updating code like this

function observePostViews($postID) {
    $count_key = 'post_views_count';
    $count = get_post_meta($postID, $count_key, true);
    if(!isset($_SESSION['posts_viewed']) || (isset($_SESSION['posts_viewed']) && !in_array($postID, $_SESSION['posts_viewed'])))
    {
        if($count==''){
            $count = 0;
            delete_post_meta($postID, $count_key);
            add_post_meta($postID, $count_key, '0');
            set_session_posts_viewed($postID);
        }else{
            $count++;
            update_post_meta($postID, $count_key, $count);
            set_session_posts_viewed($postID);
        }
    }
}

Hope it fix the issue. Let me know if you have trouble.