The good news is that there is another way to shut off the water using something called the angle stops that are on the bottom of the sink or if you have a cabinet they will be inside the cabinet.

The angle stops are usually gold, chrome or rough brass in color and what you will need to do is to shit off the angle stops by turning the valves as fast as you can to thr right ( clockwise) to get the water to stop instantly.

Once and a while there will be a valve that will be "frozen" or stuck and will not be able to be shut off and this is when you will have to find the water main to the house and shut off all the water to the entire house and then have to call the plumber to replace the valve as well as fix the original plumbing problem you had from the start.