The government are only responsible for a small amount of new fiat which enters circulation; the banks are responsible for the majority by issuing loans under the fractional reserve system.

Let's say you deposit $100 with your bank. The bank credits your account with $100, and then uses your money to loan out to other customs, keeping a "fraction" of it in "reserve" (lets say 10% for the purposes of this example). The bank keeps $10 in reserve, and loans out $90 to customer A. Customer A spends that $90 with a business, which then deposits $90 in to their account. The bank moves $9 to its reserves, and loans out the remaining $81 to customer B. Customer B spends that $81 with a business, which deposits it to their account. The bank moves $8.10 to their reserves, and loans out the remaining $72.90. And so forth. Take this down to its conclusions, and the total amount of new money entering circulation based off your original $100 deposit it $1,000, ten times as much. This is known as the "money multiplier".

This is the fractional reserve system. The banks only ever have a fraction of all their account balances in reserve, and why a "bank run" is possible; too many people trying to withdraw their money all at once can cause the bank to collapse as it quickly runs out of reserves.