Using the simple app.run() from within Flask creates a single synchronous server on a single thread capable of serving only one client at a time. It is intended for use in controlled environments with low demand (i.e. development, debugging) for exactly this reason.
Spawning threads and managing them yourself is probably not going to get you very far either, because of the Python GIL.
That said, you do still have some good options. Gunicorn is a solid, easy-to-use WSGI server that will let you spawn multiple workers (separate processes, so no GIL worries), and even comes with asynchronous workers that will speed up your app (and make it more secure) with little to no work on your part (especially with Flask).
Still, even Gunicorn should probably not be directly publicly exposed. In production, it should be used behind a more robust HTTP server; nginx tends to go well with Gunicorn and Flask.