I've played three games so far (won all three).
Shoshone (Cultural, Prince, Turn 381), Poland (Science, King, Turn 377), Venice (Diplomatic, King, Turn 382)
It's crazy they're all within five turns! I used to play G&K on both King and Emperor and usually won between turns 330-370 so I'm slightly slower than that now. My fastest victory ever has been on turn 310 (I play large or huge maps and I'm not a pro player). Next I'll try Emperor and see if I can still win and if it's within the same time frame.