Kathmandu Valley Police spokesman Shekhar Khanal said 17 people were killed in the capital.
"Tear gas and water cannons were used after the protesters breached into the restricted area," Khanal told the AFP news agency.
Ranjana Nepal, an official from one hospital which received many of the injured, said tear gas also entered the hospital, making it difficult for doctors to work.
"I have never seen such a disturbing situation at the hospital," she told AFP.
A Kathmandu district office spokesperson said a curfew was imposed around areas including the parliament building after protesters attempted to enter.
Two were also killed in the eastern city of Itahari while protesting after the curfew order was announced, local police said.
Nepal Army Spokesman Rajaram Basnet told the BBC that a small unit of soldiers had been deployed in the streets following the introduction of the curfew.
On Monday evening, Home Minister Ramesh Lekhak resigned during a cabinet meeting, local newspaper The Kathmandu Post reported.
A minister present at the meeting said Lekhak stepped down on moral grounds following the deaths at protests, according to the newspaper.
The United Nations Human Rights Office has called for a"prompt and transparent investigation" into the deaths and urged the government to reconsider its measures for regulating social media.
In a statement, the office's spokesperson Ravina Shamdasani said the UN had received"several deeply worrying allegations of unnecessary or disproportionate use of force by security forces" during protests.
Last week, authorities ordered the blocking of 26 social media platforms for not complying with a deadline to register with Nepal's ministry of communication and information technology.
Since Friday, users have experienced difficulty in accessing the platforms, though some are using VPNs to get around the ban. So far, two platforms have been reactivated after registering with the ministry following the ban.
Nepal's government has argued it is not banning social media but trying to bring them in line with Nepali law.