How do you manage with the space crunch early in the game especially when you don't have much GBs or special buildings and depend solely on houses ?
Getting useful GBs early on is essential, in my opinion. Get a Tower of Babel early on, and boom, you no longer need houses for population. Get a Chateau, that will allow you to get rid of all the goods buildings, plus you'll get a lot of gold. Income solved. Get a Himeji Castle, and you're all set with supplies production as well.
You don't need a lot of space. I have all the GBs I want built, and still have plenty of room for non-essential stuff like wishing wells. I even have GBs that take up a lot of space and are totally useless, but I don't want to delete them.
Space is not an issue if you plan ahead.
I have to add that doing all these is a lot easier in a guild that has these resources available.