Tweetbot puts a banner across the top showing the number of unread tweets since your last visit. It counts down as you scroll toward the top. Only thing is, it's a paid app and Tweetbot 4 is like 10 bucks.
I really only read what I miss when I'm using Tweetbot. Problem I have though is that I don't get tweet notifications unless I subscribe via text and that can be hit and miss.
I really am not a fan of how the official Twitter app forces timeline organization into showing the tweets they think you'd find important instead of displaying in chronological order.
I hate that (and how Instagram does it, which is why I hardly ever use that anymore), and I hate how it doesn't resume where you last stopped reading. Having the countdown at the top is nice in the app, but I would like to have the red notification dot to tell me how many new tweets I have in my timeline.