Azure Functions - run long operation in another thread

This pattern is not recommended (or supported) in Azure Functions. Particularly when running in the consumption plan, since the runtime won't be able to accurately manage your function's lifetime and will eventually shutdown your service.

One of the recommended (and widely used) patterns here would be to queue up this work to be processed by another function, listening on that queue, and return the response to the client right away.

With this approach, you accomplish essentially the same thing, where the actual processing will be done asynchronously, but in a reliable and efficient way (benefiting from automatic scaling to properly handle increased loads, if needed)

Do keep in mind that, when using the consumption plan, there's a function timeout of 5 minutes. If the processing is expected to take longer, you'd need to run your function on a dedicated plan with AlwaysOn enabled.

Your solution of running the background work inside the Azure Function is wrong like you suspected. You need a 2nd service that is designed to run these long running tasks. Here is documentation to Micosoft's best practices on azure for doing background jobs.