How to Manage increasing size of sqlite in Mobile App?

I do not know how large is your data size is. However, I think it should not be a problem storing reasonably large data into the internal memory of an application. The internal memory is shared among all applications and hence it can grow until the storage getting filled.

In my opinion, the main problem here is the query time if you do not have the proper indexing to your database tables. Otherwise, keeping the databases in your internal storage is completely fine and I think you do not have to be worried about the amount of data which can be stored in the internal storage of an application as the newer Android devices provide better storage capability.

Hence, if your database is really big, which does not fit into the internal memory, you might consider having the data only which is being used frequently and delete otherwise. This highly depends on the use case of your application.

In one of the applications that I developed, I stored some large databases in the external memory and copied them into the internal memory whenever it was necessary. Copying the database from external storage into internal storage took some time (few seconds) though. However, once the database got copied I could run queries efficiently.

Let me know if you need any help or clarification for some points. I hope that helps you.

For max size databases. AFAIK You don't want to loose what's on the device and force a reload.

Ensure you don't drop the database with each new release of your app when a simple alter table add column will work.

What you do archive and remove from the device give the user a way to load it in the background.

There might be some Apps / databases where you can find a documentation, but probably this case is limited and an exception.
So to know exactly what's going on you need to create some snapshots of the databases. You can start with that of one app only, or do it directly with several, but without analyzing you won't get a reliable statement.
The reasons might be even different for each app as databases and app-features differ naturally too.

Faster growth in size than amount of incoming content might be related to cache-tables or indexing for searches, but perhaps there exist other reasons too. Without verification and some important basic-info about it, it's impossible to tell you a detailed reason.
It's possible that table-names of a database give already some hints, but if tablenames or even fields just use meaningless strings, then you've to analyze the data inside including the changes between snapshots.