I am working on an application where file-uploads happen often, and can be pretty big in size.
Those files are being uploaded to Web API, which will then get the stream
I think a better approach is for you to go directly to Azure Blob Storage from your client. By leveraging the CORS support in Azure Storage you eliminate load on your Web API server resulting in better overall scale for your application.
Basically, you will create a Shared Access Signature (SAS) URL that your client can use to upload the file directly to Azure storage. For security reasons, it is recommended that you limit the time period for which the SAS is valid. Best practices guidance for generating the SAS URL is available here.
For your specific scenario check out this blog from the Azure Storage team where they discuss using CORS and SAS for this exact scenario. There is also a sample application so this should give you everything you need.