10 turns (1735 AD - 1785 AD)
So looking at everything at the start of my turns, I decided that our military strength is our #1 concern right now. Just take a look.

That is not good. I know that some CPU personalities will go after you if you are this weak relative to them, regardless if you're friends or not. I don't want Mao or Victoria getting any funny ideas, so I'm halting tech for now and going straight up cash flow in order to do some serious military upgrades. That means science and culture sliders at 0% for a few turns.
Also starting off, I cancelled production of the forge in Medina. Forges are normally great structures to build, but we definitely don't want this here. Remember, Medina is our great person farm, so we value food and health over just about everything. We also built the Globe Theatre here, meaning we don't have to worry at all about happiness. What does a forge do? It increases production (always good), DECREASES health (very bad in Medina), and increases happiness (totally inconsequential in Medina). It's really all about culture and health related buildings there now.
Building wise, didn't really accomplish much during my turns. Science went no where as I upgraded almost all our guys to riflemen. I think I made a mistake when I suggested we learn tech up to Medicine. We probably should have pumped up the culture slider to the max instead, but I guess it's kind of too late now. If you look at the stats screen, you can see Mecca is just about halfway to becoming a legendary city with 50000 culture. I'm concerned about Medina...it's only generating about 200 culture a turn while the other two cities are at about double that. We didn't do a very good job of balancing our +50% culture buildings, as Medina will only be able to have 2. We're probably going to end up having to use all of our great artists there at the end of the game. Our next couple sets of 10 turns are going to be big. We're well on our way to winning this game before anyone can build the spaceship. The tough part is going to be keeping aggro off of us. Hatty is hates us now due to the dealings I had to do with Mao to get him off our back. I thought it was more important to be friendly with Mao just because he's on the same continent. Remember, with the exception of our 3 culture cities, military is key. We have to pretty much pump out as many riflemen as we can to compensate the computer getting ahead of us in tech and being able to make stronger guys.
By the way, here's where we're at power wise after my 10 turns of military upgrades. Still not very good, but better.

Good luck. I'm anticipating this game is going to be over in about 40-50 more turns as long as we can hold off aggro.