Cloudflare can most certainly help speed up and protect your websites. However, sites that stream content should move this type of content to a subdomain in their DNS settings that we don't proxy.
If you created a subdomain, for example streams.example.com, you could then add it in your Cloudflare DNS settings by creating a CNAME DNS record called streams and making sure it is grey-clouded.
We have seen sites have performance issues because of the number of connections that streamed content causes when running through the Cloudflare proxy.
We recently introduced Cloudflare Stream, a video-on-demand platform for developers and content teams building video applications. Cloudflare Stream combines video storage, processing, global delivery, and playing into a unified video streaming solution. Learn more about Cloudflare Stream.
For Enterprise customers, Cloudflare can allow caching/streaming of media whenever it would be considered appropriate for that account. To learn more about the Cloudflare Enterprise plan, contact us.