I was playing on difficulty 3, but I thought the AI was too slow and dumb, so I switched to difficulty 4, but I think there's something wrong. I only went up 1 level of difficulty and now I'm in this situation every time I start a game. The AI darts in front of me without giving me any chance of reaching her. Is there a bug in this difficulty or is it like that?

The strange thing is that usually only 1 civilization that drops in front, the rest is very similar to me.

Yep, that is the usual behavior of the AI - one of them can dart in the first 4 Eras but then you usually catch up and blow them out of the water by Era 5. The key part is getting the Patronage tech as soon as possible and also extra territory in the New world. Or you can just wage war and get new territory from your neighbors. :)

The AI in front is either alone on a continent or managed to wage a couple of successful wars against other AI. That gives them population, territory and military stars. On Nation difficulty the AI becomes a lot more aggressive and starts to apply pressure on you if you are lagging behind.

As you increase the difficulty, the AI gets better at rushing past the opening of the game. This becomes very apparent if the AI persona gets some sort of static bonus to yields (+10 Industry on makers quarters for example.)

They will usually slow down in fame progression around the 3rd or 4th era. Their tech advancement usually also slows around then, as there are so many potential choices to make for each tech, and human players are usually a bit better at traversing the tech tree.

A general strategy to beat them is to hang back in an early era (1-3) and farm stars if you can. The viability of that is based on many factors, and if you get too greedy you end up falling behind again. Expect to reach fame parity around Early Modern or Industrial, depending on your skill level and playstyle.