As I was researching short films, I came across Flim4 Shorts: http://www.film4.com/minisite/short-films which is part of the Film 4 website which is dedicated to showcasing short films.
When you first go into the website you have the opportunity to look at a variety of different options on the website.
If you click onto the 'Film4 Productions' it takes you to films which will be showing on the Film 4 website. However they are full length films that have been produced by Film 4. To view short films, all you have to do is click on the 'Video clips' button on the banner and it will take you to a page where you have the option to watch whatever film you want. However, they have a very small amount of videos on this website as they only have 13 on at the moment. This may be because they remove short films once they have been on for a certain time period.
Here are a couple of films from the website:
http://www.film4.com/film4-productions/shorts/pitch-black-heist
http://www.film4.com/film4-productions/shorts/see-me
