Part of city management is planning for your expansions. When you get a new expansion, instead of filling it immediately, use the opportunity to rearrange your city... especially plan for the growth (changing dimensions) of the buildings you already have. I completely redesigned my city when I found out that the sizes change, and made sure to leave room for the eventual growth.
This is a city planning game, and like the puzzle talked about above, it requires planning and thought before moving things around. You can free up additional space by deleting one or two low level buildings, or decorations, while you conduct the reorganization. After I planned my city more economically, not only did I have room for my buildings to expand, but I had enough room for everything I had before, just by arranging to use the space better with fewer roads.
I would also recommend looking at the rankings and visit to see how other high level people are organizing their cities, as well as making a note of the dimensions of your "future" structures. That will help your city development to be less chaotic.