Well, if a server gives you a CORS error there is not much you can do from the client.
This is a good article on CORS that I recommend you check out, HTTP access control (CORS) - HTTP | MDN
However I would be very surprised that Twitters API has cors headers other than allow all (*), in fact they don't.
Can you share the full code? Communicating with the Twitter API client side is tricky business overall after they deprecated the 1.0 API.