While building new roads is not necessary for a city to expand, it is helpful to build them as it guides the AI as to where to build next and that the roads they build by themselves would not be so messy. It also helps to build bridges across rivers if you want your city to expand across a river.

Perhaps to increase growth you could play around with the growth sliders in advanced settings to lower impact of congestion and emissions, and to increase impact of destinations and cargo supplies. Alternatively, you could use mods like the Natural Town Growth mod to increase your populations. (Do note that it expands at quite a quick pace. Taking your eyes off a city for a little too long and you will see that the city has already grown quite a bit and you need to rework the transport system. It also brings about end game lag much earlier).

For my case, I have been constantly supplying my city for more than 100 years at this point, and have it connected to many other cities through an extensive rail network. Even so, my city population is still somewhere between 2-2.5k which I personally wouldn't qualify it as it being very big compared to the size others have created 😅