I can’t find a user manual for the hair styler so check the information label (hopefully it has one) on the styler to see if it shows the specifications i.e. voltage (V) and power (W).
Being made for the USA market it will be 120V AC but knowing the wattage (power consumption) will let you know how much current it requires.
Use Ohm’s Law to work out the current that is required by the styler. Click Clear to remove the figures already in the voltage and current boxes in the calculator and insert 230 as the voltage value and then insert the W rating value as shown on the styler. Then click on calculate
Your travel adapter/converter specs state that it is only capable of supplying 250W total over its 3 AC outlets, so at 230V this is only 1.08Amps. It may be that it is shutting down to protect itself, due to being overloaded if the styler uses more power than 250W.
What is the make and model number of the garment steamer? Does it indicate that the power is connected to it e.g. power on/off light? What voltage is it designed to work with?