Which foreach loop is not iterating as many times as you want? Is it the last one (foreach $link(@links))? What are the command line options when you run it? What are the contents of linklist.txt? When I run this with no command-line arguments and linklist.txt containing the names of two web pages, I get two runs (although what I see is '1 Empty run' printed twice, which may or may not be what you want—see below). What should I see?

Note that you almost certainly don't want to roll your own option-parser. For example, your parser would be perfectly happy to accept two (or zero) arguments to -U or -x, which is probably not what you want. Something like getopt is probably more appropriate.

Note that the quotes around $link in the call check_link("$link") are unnecessary—they would 'stringify' $link, but it's just a simple scalar read from a file anyway.

What is the significance of $response and $part? You perform substitutions on them, but you never seem to use them.

That last foreach loop is probably more appropriate as a while loop anyway: Unless you need to know in advance all the lines you'll be processing (which doesn't seem to be the case), it's much more memory-efficient to process them one at a time.

UPDATE: I don't have Switch installed, so I used Perl 5.10's given instead. Since ikegami says below that switch is the problem, that could be why I'm not seeing anything.
UPDATE 2: If you do want to keep your hand-rolled parser, note that you must anchor your regexes at the beginning of the string. /-/ will match a string containing a hyphen anywhere, not just at the beginning.