They set the machine to play at 2-ply, i.e., it only looks two moves ahead. It's not so surprising even a good algorithm would lose in 15 moves at 2-ply.
The article mentions they also played at 5-ply and it lasted 27 and 30 moves against Kasparov, which is probably better than most casual players would do.
I do think this was a reasonable choice for the presentation because as I understand it, that's what Turing would have used when working it out with pencil and paper (and as they said, even that would take 15 minute between turns). 5-ply probably would have taken on the order of weeks or months to calculate the next move by hand.
The article mentions they also played at 5-ply and it lasted 27 and 30 moves against Kasparov, which is probably better than most casual players would do.