A fuel injector is provided for a turbine engine. This fuel injector includes a fuel nozzle, and the fuel nozzle includes a gallery, one or more feed passages and a plurality of exit passages. The gallery extends within the fuel nozzle circumferentially around an axis between a first end of the gallery and a second end of the gallery. A size of the gallery changes as the gallery extends circumferentially around the axis between the first end of the gallery and the second end of the gallery. The one or more feed passages extend within the fuel nozzle to the gallery. The one or more feed passages are configured to supply fuel to the gallery. The exit passages extend within the fuel nozzle from the gallery. The exit passages are configured to receive the fuel from the gallery.