You could speed things up like this:

#!/usr/bin/perl

use strict;
use warnings;
use WWW::Mechanize::Firefox;

my $mechanize = WWW::Mechanize::Firefox->new( stack_depth => 0);
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Mechanize keeps a history of every page with its state; also, it makes a clone of each Mech object. If set to 0, it won't keep that history. You can adjust it to meet your needs.