Hey! I think I've found a solution to this problem.
I'm sure you've done your fair share of scouring the internet for solutions, but like me didn't have any luck finding a proper solution. I decided to go tinkering around in the settings for the app - I'm using the CurseForge launcher now, though I imagine at the time of writing you were using Twitch. Obviously they're transitioning over to CurseForge, so I'll proceed with what I did for this.
First off, I went into the settings of the app - little cog in the bottom left corner - and went to Addons. There's a little slider for Backup Space, which I bumped all the way up to the maximum it would let me, 10MB.
Then, I went to download the modpack. It got further than it was normally - typically it was stopping at mod 101 of 181, which was a pretty big one, but this time it got somewhere around 150 before it gave me the error again. That was frustrating.
I tried downloading it again, but this time while it was downloading, I did not do anything else in the background. No YouTube videos, no Discord chats, nothing that would use up broadband. I just let CurseForge download and install the modpack. This time it worked.
Tl;dr: Go to settings, addons, set backup space to max. Go to download your modpack and don't do anything else online while it's downloading.
I was having this problem repeatedly. I finally got it to work by doing the following in Curseforge Settings > set backup space and simultaneous downloads to max Settings > Minecraft set default memory allocation 9gb < - I don't think this setting had an effect. I'm just including it for completion as I did change that too.
So just a heads up to all that see this, increasing the backup storage space in the curse app doesn’t actually do anything for Minecraft. It’s a wow feature... not a Minecraft thing. Most it will do is increase any wow backups made, which won’t help at all with this error. . .
The real issue with the WebClien Request is actually just a simple network issue. Either your internet isn’t strong enough or is unstable at the time of download. Easy fixes:
1: Pausing or closing out anything that is using the internet other then the app, this includes: YouTube, other downloads, streams, etc.
2: If your using wifi try a wired connection or get closer to the source if you can’t do a wire.
3: Reboot your pc, can help close programs running in the background.
4:Restart your router, like computer they can benefit from a reboot from time to time.
5: If all else fails try again in an hour or so, if the network where you live is busy try it a a quieter time.
if nothing works at all try asking on the Curseforge discord
I'm guessing it's fixed by now, if not or for future readers. Make sure you have enough disc space and preferably have the path of installation set to your D: drive which is normally the storage drive which isn't used for computing operations by the pc unlike the C: drive. This can be done by changing the path in CurseForge by going to "Settings" "Addons" and then at the top it reads the path with a button labelled "change" next to it. Just link it to a folder on your D: drive.
NOTE: changing the path might result in modpacks downloaded before the pathchange to not function anymore as it can't find them due to them using the old path.