I disagree with most of your post. There really isn't that much space to build, because the extra tile unlocker is actually useless. The game can't even run the simulation for 9 tiles, let alone 81, it runs out of assets somewhere around 7 tiles.
What? So how does squeezing your city together help your game if you claim you can't fill all 81 tiles anyway? That literally makes no sense.
Further, real cities use grids. Nearly every modern city is based on a grid design, with deviations for natural terrain. Only in cases where cities were large and developed pre-industrial revolution do you get crazy looking road designs, so this mostly effects European cities. Take a look at every major american city- it's going to be a grid unless the terrain is terrible (Seattle, San Francisco).
*snicker* That's a very, very American perspective. You should try looking outside your country to other cities to see what most road layouts are really like. Anyway, I think your city looks fine. There are grids, but at least you veer away from them in different districts and follow the river. Any traffic problems you're experiencing with this layout can be mitigated by analyzing vehicle routes and providing alternate paths. I've got a whole spaghetti network of highways for that and my traffic flow wavers between 87% and 92% with a population of 85'000.