Increasing the number of turns increases the output voltage, but decreases the available current at a much higher rate because you have to make the wire diameter smaller to get more turns on the same pole piece. The maximum flux density that a given magnetic path can support remains constant unless you add more iron or decrease the gap.
If you add turns, the voltage would be higher, but for a given load resistance, so would the current. However, since flux density is proportional to nI (n is turns, I is current) you have to reduce current to keep the flux below saturation. You have to reduce the wire diameter to fit the larger number of turns on the same pole piece. The net output power (product of voltage and current) at saturation therefore stays the same as you add turns.
There is an optimum number of turns (wire size that will fit on the pole piece) which gives the optimum voltage for a given load, like charging a battery.
Take an automotive alternator for example. The number of turns on each stator pole has been optimized to deliver the max power available at an output voltage consistent with the battery being charged (14.4V in a car). The iron in an automotive alternator is capable of delivering about 750W of output. If you tried to rewind that alternator to charge a 28V battery, you would still be getting about 750W, but the current would be about half.