There is actually a web app specifically for this - It's called Kenmei. It lets you not just keep a list of what you are reading, but it's specifically built to keep track of chapter releases on different sites, both official an unofficial.
You can find it here → Kenmei | Cross-site Manga Tracker
I'd say closest comparison to this site would be MangaUpdates, but Kenmei is a lot more modern and has a lot of features that MangaUpdates doesn't have (like Smart Suggestions).
But it's still in alpha, so it's not as comprehensive in terms of data, so I'd go with @Makoto recommendation as well - use MyAnimeList or Kitsu as a backup and place to find out more information, like synopsis and so on. But when it comes to just keeping up with latest releases of new chapters, I don't think there is anything better than Kenmei out there (at least on the web, there are mobile apps like Tachiyomi that does similar stuff
Here how it looks:
Disclosure: I am the developer for this app. After reading manga for almost 2 decades, I still felt like there was no easy and modern to use app for all the series I read. Which is why I decided to build Kenmei