Is there a way to buy city districts instead of building them the "normal" way? I can't seem to buy them with gold or faith.
You can't spend Gold, Faith etc. to speed up building a district, the only way is to build them the normal way.
But: there is at least one way to grant an instant production boost though - chopping down woods, or harvesting resources.
If you are playing as the Aztecs, you can rush them with workers, and you can buy workers. Otherwise no.
In the new Rise & Fall expansion there is a governor
Reyna - The Financier that when promoted can allow you to purchase districts out-right with gold in cities that she has been established in.