Choose a server for media website

Best hosting for a media website would be vps server. This way you can have more control over the site and how much traffic and space you need.

Look for a cheap vps hosting company that allows you to upgrade to larger hosting package free. No set up fees top upgrade. This way you can get cheap vps package and upgrade to larger one when you need to. You can upgrade to a dedicated server anytime if you need to.
Assuming you are asking for a web hosting provider who allows and manages video streaming and media efficiently, you can refer to this list

Hosting providers offering video streaming - Web Host Lookup

You can check if their offerings are meeting your requirements and then make a decision.

If however you mean something else by media hosting server provide more details so that a correct answer can be provided
VPS hosting is the best to host a media website. It supports all the media streaming contents.

I am using AccuWeb Hosting's SSD Windows VPS fully managed plan since 2015. You can know more benefits of VPS from here.