So the OP's puzzle included the count for getting to the card, but the puzzle seems far more interesting the way everyone is interpreting it anyways.
Anyways, I achieved 4 turns guaranteed! Many simple ways were very close (got 7 provinces, or ALMOST had a guarantee, or needed one more card's worth of draw power), but I couldn't get any to work. Finally, I did come up with a solution, but it's pretty complicated and uses all 10 Kingdom cards...
Here it is:
We will use the 10 Kingdom cards:
Watchtower, Develop, Bridge, Throne Room, Ironworks, Minion, Inn, Library, Border Village, King's Court
1. Buy Bridge
2. Buy King's Court
3. Play King's Court the Bridge, and play Platinum. Buy Develop, Throne Room, Border Village gaining Minion, and Border Village gaining Inn. Shuffle all the cards into your deck.
4. Play a Border Village (we necessarily draw one of them), drawing the last card in deck (discard is King's Court, Bridge, Platinum). Throne Room a Develop, turning Border Village into King's Court + Library, and then Inn into Border Village, gaining Ironworks, and Ironworks. Minion for cards, drawing Border Village, Ironworks, King's Court, Library (necessarily, since the other Ironworks goes in discard). King's Court Ironworks, for Throne Room, Watchtower, Bridge. Library, drawing Platinum (in worst case), and the rest of our actions except Bridge. Throne Room a King's Court, playing Bridge, and then Ironworks, gaining King's Court, King's Court, Bridge. Watchtower to draw these and the last Bridge. Play King's Court, King's Court, Bridge, Bridge (and Border Village).
Extending the ending, a bit, we can actually get 8 Colonies (and 3 Provinces).
4*. On what was the last Ironworks, instead take King's Court, King's Court, Ironworks. Watchtower to draw these and the last Bridge. Play King's Court on King's Court, first playing Bridge, then Ironworks for King's Court, Bridge, Bridge, and finally Border Village to draw what we just gained. King's Court a Bridge, and Bridge, making a total of 10 Bridges out (along with a useless Platinum that got us this far in the first place...)
Unfortunately, I can't provide a game log since this set of cards doesn't allow for getting my deck down to that one card.... so I sorta haven't fully solved the puzzle. Can anyone figure out how to use less cards? Or just generally figure out something simpler than what I did?
One thing that surprised me was that I didn't Governor, which I thought would be good for this. I thought King's Courting it was great for the draw, for turning the damn Inn into a King's Court, and for turning Border Villages and gained Golds into Provinces. But it wasn't working out. Maybe you can find it effective?