Although the endpoint to connect MySQL is a public address, as long as the frontend and the backend are in the same region, there will be zero latency related issues between them compared to having a private address. I'd suggest getting rid of ClearDB MySQL if that's what you're using. It's not good enough for production environments. Try MySQL as VM or the new MySQL as a Service from Microsoft.
Azure App Service does not stacks up with VMs regarding performance if you're comparing 1:1 especially if you are using Apache, Nginx or any other web server platform on the virtual machines. App Service running Windows can be very slow for Php applications because they run on IIS. If you deployed App Service on Linux, performance is better, but still not as good as running on VMs, and still in preview.
One thing you can do to improve performance is to enable Cache via WP plugin. From previous work with customers, enabling page/database cache was enough to increase load speeds to almost the same level as the virtual machines.