Why do you rely that much on the shell and external commands? That makes your code more fragile, less portable, and less secure than needed.

• You don't need wget, LWP::UserAgent or one of its subclasses can do everything that wget does, in Perl. Some people prefer WWW::Curl, that should also be able to replace wget.
• chomp(my $wget = `which wget`); assumes that there is a which utility available in $ENV{PATH}, that the which utility is friendly, and that it returns either nothing or the location of the wget utility. Not every platform has a which utility. Not every which does what you expect from it. And there may be one strange which utility that returns other data than you expect. If you just know the name of an executable, but not its exact location, search $ENV{'PATH'} in Perl. File::Which can do that for you.
• You don't need a pwd executable, it's probably not located in /bin, and it's not available on several platforms. Use the getcwd() function of Cwd.
• You don't need a mkdir executable, it's probably not located in /bin, it's not available on several platforms, and it does not always accept a -p parameter. Use the mkdir function for simple cases, and the mkpath() or make_path() functions of File::Path to replace mkdir -p.
• Your code constructing $exe assumes a bash-like shell (not available on every platform, at least not by default) and very friendly script input. You should AT LEAST use single quotes instead of double quotes to prevent shell interpolation of special characters. You should also escape quotes existing inside your quoted strings. Better, you should not rely on any shell and use the multiple-argument form of system instead. Of course, that won't allow the "cd $outdir" before calling wget, you would need to call chdir before. Best way: Don't use any external command, use available perl modules like LWP::UserAgent instead.
• Did you know that wget already has the ability to process a list of URLs from a file? Just use the -i parameter.
• Did you know that wget already has the ability to download files to a different directory than the current one? Just use the -P or --directory-prefix parameter.
• Did you know that wget already tracks the time it needs to download a file?
• Using just the wget documentation, I could replace your entire perl script with a single call to wget, except for that strange renaming thing that is probably caused by using the wrong download URLs. sed -e 's/\.\(html\|htm\|xml\)\s*$//' or perl -pe 's/\.(html|htm|xml)\s*$//;$_.="\n"', used as filters, would fix the URLs before passing them to wget -i -.

Alexander