Traditional currencies have inflation because they are printing money faster than old bills get destroyed.
The amount of "printed currency" in circulation has almost nothing to do with the size of the money supply. It's amazing how many gold/bitcoin fans don't understand this. Heck, the US recently concluded "QE", in which a couple of trillion dollars were created by the Fed without any of it being physically printed.*
When you deposit $US with a US bank, in a savings account or CD, it can loan out 100% of your deposit. If banks offered BTC-denominated savings accounts, they'd work the same way. If you're thinking "but wait, that means there wouldn't be enough bitcoins in existence to allow everyone to withdraw their deposits", then congratulations, you understand how banking works.
There's only ~$1.3 T in physical US currency, but there is ~$10 T in total bank deposits (of various kinds). That wouldn't change if we adopted gold or bitcoins as the national currency.
*To further complicate things, the money supply actually didn't grow during that time, as bank deposits with the Fed grew at about the same rate. When the banks start finding investments better than the rate the Fed pays, we'll start seeing the inflationary effects - it's a very new idea by the Fed, and time will tell how crazy it was.