Make tag archive display post are ordered by post format

You’re right. It isn’t a good structure.

1. You are running three queries on the page– the two you are creating
   plus the main query that is being completely ignored (plus ancillary
   queries).
2. You are also clobbering the main query halfway through the page load
   when you overwrite $wp_query, which can cause unexpected and
   unpredictable results and difficulty with pagination.
3. You are overwriting other Core variables, like $posts
4. You have PHP tag SPAM
5. Your code formatting in general is lacking, though I’ve seen much
   worse.

What you want to do is alter the main query to exclude that post format:

function exclude_asides_wpse_194888($qry) {
  if (is_tag()) {
    $taxq = array(
        array(                
            'taxonomy' => 'post_format',
            'field' => 'slug',
            'terms' => array( 
                'post-format-aside'
              ),
            'operator' => '!='
        )
    )
    $qry->set('tax_query',$taxq);
  }
}
add_action('pre_get_posts','exclude_asides_wpse_194888');

Then run your “asides” query:

$asides = new WP_Query( 
  array(
    'posts_per_page' => 4,
    'post_type' => 'post',
    'tax_query' => array(
        array(                
            'taxonomy' => 'post_format',
            'field' => 'slug',
            'terms' => array( 
                'post-format-aside'
              ),
        )
    )
  ) 
);

if ( $asides->have_posts() ) {
  while ( $asides->have_posts() ) {
    $asides->the_post(); ?>
    <div class="col-sm-3 col-xs-6">
        <?php get_template_part( 'content', get_post_format() ); ?>
    </div><?php 
  }
}

Then let the main query as normal:

if ( have_posts() ) {
  while ( have_posts() ) {
    $query->the_post();
      if($i%4==0) { ?>
        <div class="row"><?php 
      } ?>

      <div class="col-sm-3 col-xs-6"><?php   
        get_template_part( 'content', get_post_format() ); ?>
      </div><?php 
      $i++;
      if($i%4==0) { // if counter is multiple of 3, put an closing div ?>
        </div><?php 
      } 
  }
  if($i%4!=0) { // put closing div here if loop is not exactly a multiple of 3 ?>
    </div><?php 
  } ?>
  <div class="blog">
    <div class="row">
      <div class="col-md-12"><?php 
        posts_nav_link(); ?>
      </div>
    </div>
  </div>
}