One big performance improvement you can make to your websites is to use a CDN (Content Delivery Network). This ensures that files such as images, videos and website assets are sent from servers closest to your website visitors.
Microsoft have made it very simple to create your own CDN on Azure. This is covered in detail in my post on high performance image processing with Image Resizer and Azure.
One thing that you may need to do is to serve your CDN assets over HTTPS. To do this you'll need to enable HTTPS within the Azure Portal:
This will provide you with a HTTPS endpoint e.g.
You can't yet use your own domain/SSL certificate with Azure CDN although there is a big request for this on the Azure Feedback Site.
If you've created a CDN for an Azure Cloud Service (e.g. http://[yourapp].cloudapp.net/cdn/) it's important that you create a self-signed certificate for your Azure domain ([yourapp].cloudapp.net). If you're using Azure Virtual Machines this can be done through IIS.
This caught us out recently when we enabled HTTPS for an origin server that did have an SSL certificate but it was for our own custom domain and not the *.cloudapp.net one.