Scaling practices turn a relational database into a non-relational database.

A few days ago I read an article on the internet where the writer claimed to get a 4000% performance improvement when not sorting data on the MySQL database, but in the PHP code. As if there are only two types of databases: relational and non-relational?

He has a point. 
A long...long time ago, there were no relational databases. There were just databases, these are nowadays called hierarchical databases. Indeed these were a lot faster (2-3X), but you had to write a lot more code to get the data out and indeed sort the data in the application (you retrieved rows one by one and had to traverse manually through the table/recordset/ database). 

A simple query like e.g. 'select bla from table a, table b where a.id = b.id order by a.id' could easily costs a few pages or procedures (sections) to write in a 3GL language.

However, 4000% improvement can't be true, I would fire the query developer or the DBA on the spot who set up the database.