Turn by turn, here's some things I might have suggested doing differently:
1+2) Buy at least one silver in your opening pair here. I see that you were engine building here with a draw card and an actions card, but ultimately your deck has to be worth drawing.
3) You've already got a crossroads and an Envoy here. You had $6 and you bought torturer. Sure, its a good card, but gold is better.
4) Same truth again
5) City, really? City is hard to make worthwhile in a game without +buy or other means to get cards at more than 1/turn. Yes, the curse pile was going to run down eventually, but the problem with relying on this is that the curse-giver is Torturer, and the curses are optional.
6+8+9) You really ought to have bought a gold by now. And on these turns especially. Just look through, and see all the times that you drew a city, if they'd been golds instead and how much greater your buying power would have been.
11) No excuse for not using the Torturer really. 4 copper plus a draw could have been a VP card.
12) Trading Post is a weak late game buy. It means you're digging upwards. turn 12 is really too late to be tuning the deck. At that stage, you should be starting on Provinces.
14) Apprentice on $7? Apprentice is again a superb card, but it needs gold to draw into.
17) City? Now? Bad play. Shoulda been a gold or a duchy.
18) A sure signt hat you've got excess drawing power, this turn. On the one turn you hit province, you have a $9 excess. Great if you have +buys. Otherwise, less drawing and more money is more consistent.
After that, your play is fine as you grab for Duchies, but its too little too late. The key weak point though? Turn 3 should have been a gold.