How can I lose 20 pounds quickly?

I am 14 years old, 5'7" and I weigh 121 pounds. It is really gross because I am so so so fat. If I make one move my fat stomache jiggles an my legs are so wide it is digesting. I am really overweight. I have been dieting for about one week now and have lost only 6 pounds. The only problem with this diet is I eat 500 calories and sometimes I get hungry. Also when I try to exercise I fell like I am about to faint. Yesterday i had only ran 3 miles and I got really dizzy and passed out. I would really like to know how I can exercise without this happening. I would also like to know any negative calories. Another problem I have is my mom thinks I look thin and she won't let me diet because she thinks it is unhealthy which makes to sense. I need to know how to hide it from her. And if I continue on this diet, how fat can I lose at least 20 pounds. Thanks for the help!!