Yeah, that's how I figured out the cause of the problem. Had fixed a load of performance problems on a page, but couldn't get rid of 500ms compared to my laptop and realised it must be query latency and was surprised to see 100 "SELECT 1"s using the same connection consistently took 300-600ms (and it jumps around all the time in that range). The ridiculous thing is that the test server running on S1 has better performance (but still has a 200ms query lag for those 100).
And that's leaving aside that heavier queries take 6 times as long on a so called "Premium" instance compared to my 2 year old laptop. Part of me suspects one of my cheap VMs running SQL Express could do better than azure's "premium" db performance.
I also know it's not because of DTUs (MS's database usage metric), my client's site is seasonal and is just ramping up to prime-time, it had the same crappy latency in July when it was handling 1/10th the traffic as it has now.