I think it's really worth considering self-hosting these days. Doesn't take much of a machine to run a web app, for instance.
I do a little of that, but for anything public I put it on a VPS.
I use one of the big providers; they're all the same. Empty linux virtual machine with an IP. That's all you need. They start at $4/month or so, maybe cheaper if you shop around.
A suitably specc'ed VPS from any provider with a good enough reputation. You can start small and upgrade if/when needed. I used to be a fan of Linode, also Hetzner but their recent price hike was unfortunate. Any provider that's been around for a while will probably have similar services.