Bypassing the firewall is nearly always a bad idea, also while troubleshooting. I would start with Captures (Wireshark/tcpdump) of the web-sessions. One capture for a working attempt and one capture with a failed attempt. Then compare these two captures if you spot something different.
Another troubleshooting: Have you tried lowering the MTU/MSS and try if the websites work with that modification? If it works, the PMTU discaovery could be broken in your environment..
If you found this post helpful, please give it Kudos. If my answer solves your problem, please click Accept as Solution so others can benefit from it.