You kept buying cities, which at this point were just acting as villages. Your opponent instead put these cities to good use by getting some rabbles, draw bigger hands, and thus was still able to catch up with your city purchases.
My first buy was a Wharf, not a City. My opponent only bought a single Rabble before the Cities kicked in. So this is just flat wrong.
The only difference between our initial play was that I chose to get a Wharf instead of a Rabble, and the one time I had a Potion and 5 coins I chose to buy a City, not a Golem.
Choosing the Golem over the City was probably a mistake, although the more I look at the transcript the more I think I mostly got unlucky. I drew 4 coins and a Potion on Turn 3, when I had to chose between University and Golem.