Two notes:

Instead of using @visited_urls you might be better of using %visited_urls, so you can avoid constructs as

  my @visited_urls = ($url_name);
  ...
  while (@website_links) { 
    if ((grep {$_ eq $website_links[0] } @visited_urls) > 0)
[download]

and use this instead:

  my %visited_urls = ($url_name => 1);
  ...
  while (@website_links) { 
    if ( $visited_urls{$website_links[0]} )
[download]

Also, as we don't see the rest of your code, ensure you have a sleep command in your main loop to avoid hammering the webserver.