Security officials said on Monday that three children were among the wounded.
No group has claimed responsibility for the explosion, which damaged the school as well as dozens of shops, houses and vehicles.
It was the worst incident of separatist violence in Kashmir since moderate separatist politicians began a historic trip to Pakistan on 4 June for talks on the disputed Himalayan region, which is claimed by India and Pakistan.
"I have never witnessed such a devastating blast. We are still waiting for details," Abdul Rashid, a police official, said.
Police fired in the air and used tear gas to disperse hundreds of people who blamed the government for the blast and were holding demonstrations.
"The people are grief-stricken and angry, but it is confirmed that the blast was carried out by militants," Sheikh Owais, a senior police official, said.
"I have never witnessed such a devastating blast"
Abdul Rashid, police official
Soldiers patrolled the streets after the explosion in Pulwama town, which has witnessed much bloodshed in the past 16 years.
Bomb blasts and gun battles occur almost daily between separatists and Indian troops, despite an 18-month peace process between nuclear rivals India and Pakistan, which have fought two wars over the region.
More than 45,000 people have died in Kashmir since a revolt began against Indian rule in 1989.