If you know a snake is poisonous, why would you mess with him.
If you know the Web site has security problems, why would you go to it?
You're probably not authorized to make security changes to the site, so you'll always be putting your computer at risk when you go to it. Your computer, every computer on your network, and every computer you exchange media with (CDs, pen drives, floppies, etc.)
Don't be fooled by a false sense of security with firewalls. Free software firewalls are all crap, in my opinion, because you get what you pay for. Even if you purchase a software firewall, you still have to allow the unsecure traffic into your computer before you analyze it. Is that really secure?
Hardware firewalls can be anywhere from easy to nearly impossible to set up, but even they won't stop malicious code unless you've specifically set them up to support intrusion detection, and then only if you have the proper license.
And...after all that...
If you request data from another computer, then the firewall will be expecting something to come from that site. Unless you've taken the time to configure it with intrusion detection, the firewall probably won't see any malicious code passing through it...software or hardware.
So, the question is...why do you insist on going to a Web site that, by your own admittance, is unsecure?