The key to big cities are workers:
1. If a city is growing or still can grow, you need irrigation to get more surplus food and speed up growth. If the city has grown up (say size 12 pre-sanitation) you want little, if at all, surplus food. So, you can change your irrigations to mines.
2. Store your city growth in workers. An ideal city to build a worker is at size 12 with a full food storage box. If you build a worker here, the city will drop to size 11 for one turn only and regrow immediately. When you are building your railroad network, you will have plenty of these cities and of course plenty of workers. Add some of them to cites once hopitals are available.