From your discription, your city is not getting to the correct size quick enough.
Increase your growth rate in your city. Build some buildings that increase growth. You start with one building and another is achieved with the tech civics.
You can also increase your prestige. Prestige also increases growth for your cities empire wide.
Growth per city = Prestige / # of cities. Do note that with this equation, multiple cities will slow individual city growth as the growth is spread out to each city evenly.
Best way to increase prestige is to level your sovereign. As the sovereign levels, so does prestige also increase. The tech administration also increases prestige by 1. You can also select the path of governor at lvl 4 for some champions. The promotion increases your prestige by 1. With the higher growth rate, your city will get to lvl 3 faster.
About the AI cheating, I do know that their is a bug where the AI sovereigns sometimes start with a few extra paths at the beginning of the game. That would make them easier to level their sovereign and increase their prestige for city growth.