You definitely want to check (a) the rating of the transformer and (b) the rating of your hair straightener. I think you'll find that the second exceeds the first!
Hair straighteners, dryers or anything with a heating element draw a lot of power. My hairdryer is 1800 watt and can draw something like 15 amps upon start-up. That's a lot!
We lost power during hurricane Ike last year and were running a generator for lights, the fridge, etc. I couldn't believe how I could hear that thing strain when I plugged in the toaster! Shocking how much power it takes when something has a heating element.
Yes, anything which involves generating a substantial amount of heat involves much more power than just running a couple of lights or a radio/TV/similar device.
In over-simplified terms, watts = volts x amps, or by rearranging the formula, amps = watts / volts. Hence 1800W / 120V = 15A.
It becomes more complicated when things like motors are involved, but for basic heating elements that holds true. So a 1200-watt toaster, for example, will draw 1200W / 120V = 10A. Or if it's a 1200W toaster designed for 240V operation in Britain, 1200W / 240V = 5A.