Azure App Service S2 network bandwidth

Since you're using a web role, I would assume that it is connected to your VNet by means of using a VPN gateway and a Point-to-Site tunnel, as stated here: https://azure.microsoft.com/en-us/blog/azure-websites-virtual-network-integration/

Here's the problem, the published information from Microsoft about their network speeds are as follow:

1. They never reveled the speed of their inter VNet communication.
2. They never reveled the speed of their Point-to-Site speed assuming that you're connected from a web app.
3. The only piece of information you can live with is that the VPN gateway throughput is 100 Mbps, said here: https://azure.microsoft.com/en-us/documentation/articles/vpn-gateway-about-vpngateways/

Now, since (again assuming) that your app is connected using a Point-to-Site connection, the Azure documentation doesn't give any clue or SLA to how much bandwidth you should be getting, or if there is any way to improve the speed of your communication as stated in the Azure VPN FAQ from here: https://azure.microsoft.com/en-us/documentation/articles/vpn-gateway-vpn-faq/

Q: How much throughput can I expect through site-to-site or point-to-site connections?

A: It's difficult to maintain the exact throughput of the VPN tunnels. IPsec and SSTP are crypto-heavy VPN protocols. Throughput is also limited by the latency and bandwidth between your premises and the Internet.

The only possible way for this to be answered is for you to ask the Azure support team directly.

To me, that sounds high as well, with Azure your in a shared pool environment. I bet your back-end disk is getting hammered by other clients. Based on other posts it sounds like Azure is known for this. I would see if they can relocate your backend disk to less used storage or try the recommendations in these posts or others.

Performance azure disks, high average queue length

Azure IO performance