Search for 'velocity of money'. It's a Fed chart and shows what happens when every citizen, business and government is swimming in debt. You printed all that money and gave it to the banks, but you didn't relieve the debt logjam. Oops.
Velocity of money doesn't have much to do with debt; money is a medium for negotiating the exchange of value between economic actors, the transmission fluid, as it were. If you were happy to get paid in very small increments very regularly, and similarly pay for other products in very small increments at very high speed, the economy could function with a numerically small amount of money, moving very quickly. Since that's not practical, we need lots of slack, buffer as it were, for converting our labour into money, before we in turn consume that money by exchanging it for other goods and services. There's a direct tradeoff between velocity of money and the supply of money; you need less of the latter when the former is high.
Debt is about moving around consumption and production relative to one another. It's inherently based on predictions about the future, which is why it's risky.
"Printing money and giving it to banks" is not how it works either. Rather, the government prints currency and buys debt with it. The debt represents future money, and is an asset today. But the currency isn't just given away.
The relationship, such that it exists, is thus indirect; buying debt increases money supply which makes up for low velocity of money.
