I have a site which is a white label (Multiple versions of the same site) which I\'ve launched recently. There isn\'t a great deal of traffic yet - mainly bots but probably 800
I wrote this question a couple of years ago and got notified of a minor change to the title. Having experienced more of Azure SQL Databases, I do now know the answer to this problem. For the benefit of others, it is simply that your database is set to a tier that is too low.
Azure has pricing tiers that have quite dramatic differences in performance. In order to achieve that, they throttle a lot of performance metrics, e.g. CPU power, requests per minute, etc.
This means that if you're pushing over your tier, your requests will start getting queued up as the CPU power / volume of requests is too high to process. This results in timeouts and then the request limit grows as requests wait to be processed. Eventually, it gets to the point where the database essentially goes down.
My experience is that the lower database levels, such as S0 and S1, are really under-powered and shouldn't be used for anything other than development or very basic sites.
There are some great tools in the Azure portal that allow you to debug what is going on with your database, such as the CPU graphs, index advisor and query performance insights.