It's up to the app developer whether they allow moving apps to SD. Many don't, either for performance reasons (yes, I know that the idea that Facebook care about the performance of their app is a joke!), security or just because moving to SD hasn't been officially supported since 4.0 so why should they bother. I don't have the Facebook app installed on any of my devices, but can quite imagine that it doesn't allow it.
In which case you may still be able to do it, but you'll probably need to root the device to do so. There is a procedure for changing the default install location (see e.g. this discussion
here), but no guarantee that it will work for a specific app - the app could still require installation to internal storage. To force moving to SD even if the app doesn't support it will require a rooted device, a script or app (such as
Link2SD) and possibly partitioning the SD card.
Not everyone is comfortable with this level of fiddling, in which case just using a browser to access Facebook rather than the app is an alternative. The app is notorious for draining system resources, so many people recommend this anyway.
The fundamental problem here is 4GB storage: that would have been an adequate specification in 2011, but really there is no excuse for manufacturers selling devices that limited in the last 4 years or so, for the reason you've discovered. So when the time does come to replace the device please don't underestimate the storage requirement, or assume that SD can make up for it (I would avoid 8GB devices too - with half of that taken up by the system what's left there is also inadequate for the sizes of many current apps).
