If I remember correctly, every time a tweet is made, a fan-out operation is made, adding the tweet to each of the follower's timeline. I would think the app just needs to remember the position on their own timeline?

That's part of it, but not the entire solution.

http://highscalability.com/blog/2013/7/8/the-architecture-tw...

> Outliers, those with huge follower lists, are becoming a common case. Sending a tweet from a user with a lot of followers, that is with a large fanout, can be slow. Twitter tries to do it under 5 seconds, but it doesn’t always work, especially when celebrities tweet and tweet each other, which is happening more and more. One of the consequences is replies can arrive before the original tweet is received. Twitter is changing from doing all the work on writes to doing more work on reads for high value users.