Our shih Tzu is 9 weeks old and is fed purina puppy chow twice a day.. 3/4 a cup total for whole day.. Half in morning and half at dinner.. He is pooping 15-20 times a day ..like diaharrea?

If he is new to your home, then it could be related to stress (new home, new people, away from his mom and littermates).  Could also be a parasite (most puppies tend to have roundworms) and it's best to get a stool sample to your vet to get this checked out.  Good Luck!