It's a problem with Switch or your switch. If I comment out the following bit, it works fine

   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"}
   }
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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.
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