Azure Database Size Same Before and After Deleting Most of the Rows
I think this may be answered here:
https://stackoverflow.com/questions/19677883/how-to-shrink-azure-sql-server-db-18mb-of-data-charged-for-5gb-of-server-space

To summarize
You need to reindex, then wait after you have done the deletes.

I don't think azure counts the size of logs in the space used.

Also see here to check what is using the space (I have summarized below)
http://blogs.msdn.com/b/dilkushp/archive/2013/07/28/fragmentation-in-sql-azure.aspx
Once you have done the reindex to check out various sizes run the following script.

-- Size of each table in MB
select obj.name, sum(reserved_page_count) * 8.0 / 1024 as "size in MB" from sys.dm_db_partition_stats part, sys.objects obj where part.object_id = obj.object_id group by obj.name
Order By sum(reserved_page_count) desc

-- Total Database size in Meg
select sum(reserved_page_count) * 8.0 / 1024 as "size in MB" from sys.dm_db_partition_stats