It takes a looong time to get level 5 - and i think it is supposed to.

click on the city status button (the big circle in bottom center toolbar) and it will bring up city info. Look at 'city tiles' and you'll probably see you have zero tiles available. You'll need to tear something down to build more housing.

Also,  the best way to get a level 5 city is to get the world improvement 'great theatre' under the civ research line. With a pub and inn and a palace you should be able to raise your prestige high enough to get level 5 fairly quick. After that you can tear down your theatre and palace and move them to the next city.

The city only has 50 tiles, you might be out of space? If you check in the bottom left the city's level bar will tell you how much more housing you need for the next level.

look here https://forums.elementalgame.com/393849 for the city tile issue. And be sure that you have some food left to build houses. If there is no amount of food indicated in the resource pool at the top side of the screen, you won't be able to build any houses. If so, try finding more food resource tiles or research better farming and housing tech.

Play as a Royal sovereign and stack the Palace and Grand Theatre in one city.  I got over 30 pop per turn.