Showing only future posts in archive based on custom field date

The problem here is that, most probably, you store date in CF show_date in ‘YYYY-mm-dd’ format.

You have to use the same format in comparisons…

Something like this should work just fine:

$args = array(
    'post_type' => 'esitykset',
    'meta_key' => 'show_date',
    'orderby' => 'meta_value',
    'order' => 'ASC',
    'meta_compare' =>'>=',
    'meta_value' => date('Y-m-d'),
    'posts_per_page' => 20,
    'paged' => get_query_var('paged')
);