Some will, the display backlight needs power after all and a fair amount. I believe most displays have either an internal battery, maybe some can draw power from the laptop. The former will have negligible impact on battery as the computer only needs to render the image for the second screen and send the video output the display, the latter will have a big impact on battery as on top of all that, had to power a big bright backlight.
Can you ping IPs (like your router or 1.1.1.1)? Do you have the same behavior over ethernet?
Also, might be silly bit I’ve been fooled by this more once, have you tested other devices have access to the internet and the specific sites you’re testing with aren’t down?