Wow, ok so I think I've got the worst score here(by 1 turn by the looks of it!)
I beat you for the worse score by one turn.

Game status: Science Victory
Game date: 2004AD
Turns played: 424
Base score: 1236
Final score: 1471
Time played: 12:24:00
My first GOTM and my second game of Civ5. My first game of Civ5 was played on Prince difficulty where I won with a science victory in 1987 with the Incan Empire. So an increase in difficulty of one level cost me an extra seventeen turns.
I had only one military unit and six cities most of the game, which was the strategy I used in my first game. While on Prince difficulty the AI never declared war on me; however on King difficulty in the GOTM, Austria and Sweden both declared war on me at the same time.
Austria was the leading AI civ, but after declaring war never attacked me. Sweden however quickly took two of my six cities that were on his main island, while I built up a military. This was my first taste of combat versus an AI civ in Civ5. And I learned quite a bit.
I lost quite a few military units. First Sweden's navy sunk my embarked units. Then their bombers sunk my naval units. Finally, I figured out that I need to control the skies first, then the sea second, and then I could launch my invasion to retake my cities.
Eventually after I figured out how combat worked, I regained my two lost cities and also took all of Sweden except for their capital. After the long war ended, it was an easy path to victory.