According to your description, I followed the steps below to get started with the New Relic Agent on Azure. And I could get the performance data of my Azure Web App on the dashboard of the New Relic website. You could refer to it and try to check with your steps.
1.Create a New Relic account in Azure Portal
2.Follow this tutorial to install the New Relic's agent for .NET with Portal Azure Web Apps. Also, if your application runs on Azure Classic Portal, you could follow this tutorial to install New Relic's .NET agent.
Note: Your app need to receive traffic before you could view the performance in New Relic. And it would take a few minutes for data to appear in the dashboard.
In order to check your app's performance, you could click the Visit New Relic
from the top of your New Relic account blade in new Azure Portal as follows:
In summary, please make sure that you have correctly added the configuration key values with your web app and disable the Always On
setting if you are using Azure's Standard or Basic tiers as the official tutorial mentioned. For more details, you could follow Azure Troubleshooting.