@Thomas Cerqueira Logging in Azure Functions happens in the runtime, which is a C# ASP.NET Application of its own. JavaScript Functions (and other non-C# languages) run in a worker process that runs alongside the core runtime.
The runtime is already instrumented to push metrics and logs into Azure Application Insights once configured, and with it you get the view that you have shared. This is the recommended approach for most function apps.
But you are still free to use your own logging mechanisms as required. In the case of winston, the current setup you have writes to files which are not persisted.
Instead, you should use a transport like this one (not an official Microsoft package FYI) which pushes the logs to Azure Application Insights for example. If you have other logging backends in place, you will have to use a transport built for that backend.
While I would not recommend it, if you still require logging to files, you could consider using a Linux function app and mount file shares to it, and direct logs to there.