Multithreaded use is not supported. SQLite is not a replacement for Postgres—Postgres supports many advanced data structures (lists!), SQL commands, and is faster. But in certain cases, it is enough, and of course setting up SQLite is much easier.
To be more precise, it's perfectly legal to use SQLite from multiple threads or processes at the same time. It will use its own mutexes and OS file-locking operations to ensure that at any given time, a database is being accessed by either a single writer, or an arbitrary number of readers.
In my opinion, the bigger issue is that clients can only safely access a SQLite database if it exists on a local filesystem. As soon as you decide you want to run multiple web servers, you're in trouble.
It saved me quite some hassle and reduced the attack surface compared to MySQL.
FWIW I never bumped into speed issues. My biggest issue against it is that it would let me alter columns directly which effectively sabotaged my use of alembic when I started using Python/flask.