One obvious problem with your code is that you open LINKS for appending, and then you try to read from it.

As far as I can see from a simple test, perl -e 'open(BLA,">>/tmp/bla"); while(<BLA>){print}'

The test shows that won't give you an error messages, but it will exit the loop immediately.

Which is why your webcrawler exits prematurely, without even reaching the calls to links. You should read from a different file than you are writing to.

(And you should step through it in debugger to see if what you think is happening is actualy happening.)