I went through the whole process to get to a level five city. I started with one orchard tile, one fertile tile, a wheat tile, a mine tile, and an elemental shard tile.
From past experience I knew cities needed a lot of houses in order to expand. If you're having trouble hitting the max size city, try the following steps.
Step One: I built a farm and filled the rest of the space with houses.
Step Two: I built the orchard, a town hall, and filled the rest of the space with houses.
Step Three: I built a shard shrine and filled the rest of the spaces. I got stuck here and needed to destroy the town hall.
Step Four is where I got really stuck and just couldn't expand any further. No matter how much housing tech I researched, it just didn't make it any better. By chance I destroyed a few of my houses and all of a sudden I had a much bigger building list. It turns out you need more than four spaces open to see the four-space size housing units. In order to get to the last stage, I realized I would have to destroy all of my existing housing except for a couple and turn them into the new four-tile Apartment building. Houses would take 30 people: the four tile Apartment would do 160 people. Much better. I had the option of estates, but didn't think it was worth it. They needed four tiles and only did 80 people. Worse density than the base houses.
Step Five: After I got to the max city size I had access to seven new tiles for building things. This was the first time in the city where I actually had room to build things that weren't houses or resource tiles and still expand. I did a palace and a few misc buildings. I also realized I could demolish existing housing and I still remained a size five city without losing the bonus tiles from being size five, even though I had less than a thousand people living there.
That was my experience. How about yours?
I would probably suggest that the menu communicate to me a bit better that there are improvements I could build if I had the space. Not knowing about the apartments tripped me up a bit. As well, the amount of housing seemed excessive and step four was mostly a bunch of microing and demolishing old housing to turn it into apartments in order to expand. It might help if instead of declaring the precise building we want, we could just declare certain regions "Housing Districts" and the quality of the buildings would be dependant on the tech and local amenities like pubs, restaurants, etc, that are in proximity. We could even specify low and high density districts, or rich districts, which would receive various bonuses or penalties.
I also suggest that perhaps the resource tiles shouldn't take so many tiles to get. I never bothered with the mine tile and don't think I could afford it without giving up the shard. Perhaps that is meant to be a tactical decision, but I found myself quite limited.
Lastly the population counter seemed largely irrelevant. As soon as I had the prerequisite stuff, I would skyrocket forward to the next level in a few short turns. If I didn't, the city just stagnated and went no where.
And that's my impression. Overall it was still pretty fun to build up to a nice big city.