This suggests your computer is blocking traffic to the extension sub domain for whatever reason.
Which is https://{ClientID}.ext-twitch.tv
So it’s either a computer problem or network problem your side. (I know a lot of people panic moded and blocked all extensions recently due to the IP Grabbing stuff).
And since it’s first party side not much we can do to help you here (on this forum) since it’s gonna need looking at environmentally or badgering Twitch support.
Pop open network tools and trace the problem. Check where https://es7m79v91nz4xxhzljdl3rpg9m1iuz.ext-twitch.tv/ resovles to for example.
I’m getting, currently:
ping es7m79v91nz4xxhzljdl3rpg9m1iuz.ext-twitch.tv
PING dxtkri2c61io4.cloudfront.net (99.86.119.33): 56 data bytes
In before someone threw a hosts file rule in somewhere on your machine/DNS Provider to blackhole the extensions domain.