The accident happened Friday night when their open truck smashed into a car on the road between the tiny kingdom's two main cities, Mbabane and Manzini, en route to the traditional Reed Dance, activists said.
"A total of 38 young girls have been pronounced dead, with more than 20 others seriously injured," said Lucky Lukhele, a spokesman for the Swaziland Solidarity Network which campaigns for democracy in the landlocked kingdom within South Africa.
"The girls were in an open truck which hit a sedan car stopped on the road," Lukhele told AFP.
The Reed Dance, due to be held Monday, is a beauty pageant that attracts tens of thousands of young virgins who dance before the polygamist king, who can select one of them as a new wife.
Mswati, Africa's last absolute monarch, chose his 14th wife at the celebration in 2013.
Mswati said the girls' deaths were a "tragedy in the nation".
"I would like to assure the parents who have lost their loved ones that the nation will support them through and through," said the monarch, speaking on the sidelines of a trade fair in Manzini.
"Also those in hospitals, should the need arise for further treatment they will be taken to other hospitals to ensure no further loss of life."
Police confirmed there had been "fatalities" but declined to specify the toll, with police spokesman Khulekani Mamba saying: "We won't be giving out any information because the maidens were on royal duty, so there are certain protocols to be followed before such information can be divulged to the public."
Local media gave varying tolls on Saturday morning, with government newspaper The Observer reporting hundreds of injuries and the Swazi News saying seven people had been killed, six of them girls.
News of the accident has not been broadcast on public television and journalists were blocked from photographing the crash scene, a local journalist told AFP.
Mswati III has ruled Swaziland as an absolute monarch since his father's death in 1982.