switch ($i) {
      case /-U.*/ {print LOG "Adding user to curl\n";$curla.=$i." ";}
      case /-x.*/ {print LOG "Using proxy for curl\n";$curla.=$i." ";}
      else {print LOG "Unknown argument: $i\n"}
   }
[download]

Please refer to the BUGS section of the Switch documentation:

There are undoubtedly serious bugs lurking somewhere in code this funky

Update: And here's the immediate cause of the bottom loop existing after one pass:

$ perl -MO=Deparse 707299.pl
...
foreach $link (@links) {
    chomp $link;
    $response = check_link("$link");
    $part = $response;
    $part =~ s/\r//g;
    $part =~ s/\n//g;
    print "1 Empty run\n";
}
continue {               <------ Added by Switch.
    last;                <------ Causes "for @links" to
}                        <------ exit after first pass.
[download]

First thing to do would be to ditch the Switch.

This is a deprecated module in production code, and only served to examine and explore what a real switch construct would look like. This lay the groundwork for the switch statement in Perl 6, which in turn was backported to Perl 5 in the 5.10.0 release (via a feature pragma). The Switch module is implemented via a source filter, and is the cause of many strange action-at-a-distance errors.

Your handling of @ARGV would be handled in a more orthodox manner with Getopt::Std or Getopt::Long. Doing so just might make your problem go away.

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.

sub check_link {
[download]

You didn't specify which foreach you are having trouble with. Is it this one?

foreach $link(@links) {
[download]

Also, start printing things out and using Data::Dumper.

• you might like to quote the link that you give to curl, e.g. my $curlc = "curl $curla '$link'";
• you might like to redirect the error-stream of curl to /dev/null or to STDIN (2>&1), but be aware that curl can print progress information to STDERR
• A timout of 10.000s is quite high.
• Update: If you want to check accessibility and document-existence only, it would be sufficient to download the header information only (curl switch: --head), not the whole file. Then check for /HTTP\S+\s+(\d+)\s+/ and $1 eq 200 for an existing file.