If the Fed has an interest-rate target, meaning they want to control the interest rate at a specific rate such as 4%, why will an increase in the demand for reserves lead to a rise in the money supply?

A rise in demand for reserves will shift the demand for reserves curve to the right which will cause a rise in interest rates. The Fed will then have to act to reduce this interest rate because they would prefer that it remained at the specific rate as mentioned.

To do this they will embark on Open Market Operations aimed at increasing money supply as this will reduce interest rates by increasing the supply of reserves because it will shift the supply curve for reserves to the right. The new equilibrium will be a lower interest rate.

The relevant Open Market Operation will be the buying of bonds from the public.

correct, but each time you dont pay the price of the penalty can increase.