I'm having the same issue. I never used to have this issue with downloading from itch. I suspect it's a way to force us to use their desktop app, which I managed to download and install. Problem with it is that, while downloading with it is reliable, you can't just download games, you have to install them - it auto-installs once the download is complete, and there's no option to turn this off (that I can find).
The only way to complete browser downloads for me, it would seem, is to not touch my browser until the download is complete. If I do anything, the download usually gets interrupted. Sometimes I can resume it, other times it gives me the "Forbidden" error.
They need to sort their servers out I think, but I'd happily use the app if it didn't auto-install every game.