Hit counter for multiple pages on one site

I want to use a hit counter on a links page that will record how many times each link was used. Ideally, the results would be stored into a single file (txt?)

I have used, and like, the PHP hit counter from this site. I am wondering if it can be modified to record multiple hits. This is the only idea I’ve had so far, but I don’t know PHP well enough to experiment on my own.