anonymous user, Welcome to Microsoft Q&A! Thanks for posting this question.
Just to provide you a good understanding on hosting website on Azure. Azure offers several ways to host websites: Azure App Service WebApps (PAAS solution), Virtual Machines (IAAS), Service Fabric and Cloud Services ( /- your current service). You can leverage the service that best suits your requirement.
Azure App Service (PAAS solution) is the best choice for most web apps. Deployment and management are integrated into the platform, sites can scale quickly to handle high traffic loads, and the built-in load balancing and traffic manager provide high availability. WebApp is a fully managed compute platform that is optimized for hosting websites and web applications.
Deploying an Azure Web App with Octopus Deploy behaves very similarly to the Visual Studio publish wizard and uses Web Deploy to synchronize the files in your package to the Azure Web App. Take a look at the document for more details on the process.
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Kindly treat this flowchart as a starting point. Every application has unique requirements, so use the recommendation as a starting point. Then perform a more detailed evaluation:
Decision tree for Azure compute services
Criteria for choosing an Azure compute service
Migration checklist -Migration checklist when moving to Azure App Service