Turn signal and brake light are on the DC circuit.
Without a battery installed, only the voltage that's supposed to charge the battery is on that circuit.
Since the voltage regulator doesn't have a large capacitor in or after its supply, this will be a rectified AC voltage with only a few watts of power available...
Without battery it's only to be expected that only a few bulbs at a time will work and that your blinker relay makes a buzzing/whirring sound !! Try the horn, it will sound sh**ty as well