If the leak is coming from the p-trap, turning your supplies off will not stop the leak from occurring. The supplies provide water to the faucet. So, if you turn the supplies off, the leak would continue coming out of the trap until the water in the trap has drained to below the hole/exit point.
I'm no pro, but try putting a bucket under the trap and turn the faucet on, and allow the water to run. This will ensure the trap is full of water so you can see what's going on. Wiggle the trap around with the water running and see if you see any leaks.