Hi there, I was wondering, is it possible to make a game entry on your itch page and have it forward to another website with the game? So instead of the game having an itch page or making an HTML page for it to host on itch, it would go to a website of my choice where the experience is?
I realize that sounds a bit counter-intuitive, but I'm in an experimental situation where that would be preferable.

No, but you can make it an HTML5 game and embed the other website, by uploading a web page with an iframe.

(Personally I find that deceitful and a security issue, but whatever.)

I recommend just putting a link on your itch page to the the page you want to share, to avoid confusing anyone about how the game is run.

Hope that helps