Taxonomy Overview Page?

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If all you’re trying to do is to display all post types on the taxonomy archive, then you can easily accomplish that by filtering the query at pre_get_posts. Assuming a taxonomy foobar:

function wpse111033_filter_pre_get_posts( $query ) {
    if ( $query->is_main_query() && $query->is_tax( 'foobar' ) ) {
        $query->set( 'post_type', 'any' );
    }
}
add_action( 'pre_get_posts', 'wpse111033_filter_pre_get_posts' );

This callback says “if I’m on the foobar taxonomy archive, and I’m the main query, then query all post types, rather than only the post post-type.”

Edit

Based on this comment:

Say there’s Post Type X and Post Type Y. /taxonomy_name/term1 would list Term 1’s posts, and posts in Post Type X and Post Type Y (in the main content area under the posts) that include that term.

For this, you will either need to create a custom taxonomy.php, that does what you’re trying to accomplish, or else create a custom page template.

Now say you’re on /taxonomy_name/term1/page/2 I don’t want the 2 post type loops to run and I want the taxonomy page to work as normal.

This will be tricky, but entirely possible. The process will look something like this:

  1. Create a custom taxonomy.php
  2. Use the pre_get_posts callback to include all post types
  3. Use the $paged or is_paged() conditional to determine whether you’re on the first page or a “paged” page
  4. If ! is_paged(), then you’re on the first page, so display a list of posts
  5. if is_paged(), then you’re on a page other than the first page, so offset the posts, and display a normal loop

taxonomy.php

/**
 * Simplified markup for taxonomy archive index
 */

get_header();

// First page
if ( ! is_paged() ) {
    ?>
    <ul>
    <?php
    if ( have_posts() ) : while ( have_posts() ) : the_post();
        ?>
        <li><a href="https://wordpress.stackexchange.com/questions/111033/<?php the_permalink(); ?>"><?php the_title(); ?></a></li>
        <?php
    endwhile; endif;
    ?>
    </ul>
    <?php
}
// Subsequent pages
else {
    if ( have_posts() ) : while ( have_posts() ) : the_post();
        // Normal loop output here
    endwhile; endif;
}

get_footer();

pre_get_posts

function wpse111033_filter_pre_get_posts( $query ) {
    if ( $query->is_main_query() && $query->is_tax( 'foobar' ) ) {
        // Include all post types
        $query->set( 'post_type', 'any' );
        // On the first page, 
        // display all posts
        if ( ! $query-is_paged() ) {
            $query->set( 'posts_per_page', '-1' );
        }
        // On subsequent pages,
        // offset posts
        else {
            // You could do: $query->set( 'offset', '-10' );
            // But this will break pagination,
            // so we need to fix it again
            $query->set( 'posts_per_page', '10' );
            $page_offset = ( ( $query->query_vars['paged']-1 ) * 10 ) - 10;
            $query->set( 'offset', $page_offset );
        }
    }
}
add_action( 'pre_get_posts', 'wpse111033_filter_pre_get_posts' );

found_posts

But messing with the page offset breaks the query pagination (because WordPress won’t take it into account), so we need another fix:

function wpse111033_filter_found_posts( $found_posts, $query ) {
    if ( $query->is_main_query() && $query->is_tax( 'foobar' ) ) {
        if ( $query->is_paged() ) {
            // Get the right posts
            return $found_posts + 10;
        }
    }
}
add_filter( 'found_posts', 'wpse111033_filter_found_posts', 10, 2 );