You COULD disable the watchdog for a limited period (5 seconds max, the hardware WDT kicks in and resets after this period), but you'd be better off tracking down the loop or other slow or blocking code and adding a yield() to feed the WDT and give the Module's OS a chance to do some housekeeping, process a packet, etc...
This should resolve the issue and allow for your HTML and CSS to be downloaded as separate files/requests.
Original Post:
Hi guys, RST cause 4 can be triggered by power supply issues, try putting a 10 uF (microfarad) capacitor across the power rails near the esp8266 module, this will act as a reservoir of power for sudden surges in current draw, this chip has been known to be quite thirsty and its power draw varies with CPU activity, which may be what has happened with your rapid subsequent connections from the browser to the ESP8266.
This is possibly the reason why the ESP doesn't "like" rapid subsequent requests, because the sudden demand could be seeing a surge in current draw for the CPU to service the requests.
I might be wrong, but this will help rule out power supply issues as a cause if you do this.
I'm actually prototyping with an ATX power supply at the moment because I got sick of power issues, since they have reservoir caps on all the voltage rails and it outputs an extremely stable 3.3v, which is ideal for the ESP8266.