Here's how banks counterfeit money. You deposit $100. The bank loans out $80 of your money to someone else. They put that money back in the bank. The bank now has $100 - all your money. But your checking account says $100, and the loanee's checking account says $80, for a total of $180. The bank has now effectively created--that is, counterfeited--$80 in new money. They gave this new money to themselves, and then loaned it out. Since their reserves are still over 20% (or whatever the present reserve requirement is), they keep doing this until 80% of the money is money they have counterfeited and loaned out.
And the Fed does give money to banks. They gave loans at 0% interest to Goldman Sachs who then buys government debt with it and earns interest >0%. That is the same as giving them the new money. And that is only one way, but there are others. Another way is by buying government debt from banks using freshly printed money at prices that are necessarily above what the market value would be if there wasn't an institution like the Fed that can print new money any time it wants to buy stuff.
And the Fed does give money to banks. They gave loans at 0% interest to Goldman Sachs who then buys government debt with it and earns interest >0%. That is the same as giving them the new money. And that is only one way, but there are others. Another way is by buying government debt from banks using freshly printed money at prices that are necessarily above what the market value would be if there wasn't an institution like the Fed that can print new money any time it wants to buy stuff.