I have a couple questions for you.. and they might help you see why it's not doing things the way you want.

1) Why are you using 4 different variables to refer to the same thing? You use $_, $line, $newline and $body_temp all to refer to a line from the file. Why not just use one? I'd suggest either doing

while ($line = <FILE>) {
  chomp $line;
...
[download]

or

while (<file>) {
chomp;
...
[download]

and then continue to use $_.

2) now look at where you check to see if it's a link. At the end of that, you write your new line to OUTFILE. What do you do after that? You continue to run through the code and write output again. I'd suggest either using a next, or just modifiying $line or $_ (whichever you choose to use) or writing if..elsif statements.

Hope this helps..
Rich