$title_linked =~ s|^\s+<[^>]+>|<a href="http://www.ace-installer.com">|;
will only change the first url in title linked. If you want to change all of them to relative urls you would need to change it a bit. Something like this?
$title_linked =~ s|<a href="http://www.ace-installer.com/?(.*?)">|<a href="/$1">|g;
The UK High Street