Python allows a lot of paradigms that are really difficult to JIT. I have personally seen many instances where PyPy is actually slower than CPython for a basic web app because of this.
It is only slow when you using C-Extension. Basic webapp don't use C-Extensions. If you are going to claim such please provide evidence.
We have a chat server in production written in PyPy and Tornadoweb which had sped up 20x from CPython version and test benchmarks against GoLang based Websocket with Channels and Nodejs - The pypy implementation is 30% faster than node and 10-20% faster than Golang . Big plus is PyPy version drops memory usage by 80% . CPython version have around 200MB Ram usage and pypy version only about 40MB.
On Heavy load (10k concurrent test) PyPy and Golang version are stable but Node version stop responding sometimes and packet losses occurs.
Congratulations. You sound like you have a very simple application. It is not only slow with C extensions. You would know this if you had more experience with PyPy on a real codebase.
Then define your real codebase. There is no point continuing this with just baseless talks.
There are many projects, livbs in pure python without Any c extensions. PyPy work faster with cffi based libs than python c-extension Based ones too.
Well implemented c extensions are and cffi ones are fine for PyPy. Those that have problems are ones that depends on private c-api calls
Tornado had moved to python asyncio for ages.