With API 1.1 you can achieve this using these URLs:
- https://twitter.com/[screen_name]/profile_image?size=mini
- https://twitter.com/[screen_name]/profile_image?size=normal
- https://twitter.com/[screen_name]/profile_image?size=bigger
- https://twitter.com/[screen_name]/profile_image?size=original
Official twitter documentation Profile Images and Banners
Example
https://twitter.com/TwitterEng/profile_image?size=original
will redirect to
https://pbs.twimg.com/profile_images/875168599299637248/84CkAq6s.jpg