Bitcoin doesn't have a central bank, so there's no stabilization mechanism and thus is a lot more volatile.
You're right. This is what people don't understand. They think that the fiat is stable, that's technically not and can never be. Bitcoin is however stable because it's value is directly connected to its deflationary mechanism not to any fiat currency. While fiat currencies run on artificial promises, which may seem good for today do not last forever, Bitcoin runs on natural values and the promise is based on Proof of Work which is the measurement for real value.