A police officer said that 3,000 devotees were expected at the event but 70,000 showed up.
Some of the injured people are in a serious condition in hospital. Of the 19 dead, 14 are women.
Raj Bahadur, one of the organisers, told the Press Trust of India news agency that after police started turning people back from an overcrowded bridge, a rumour spread that the bridge was broken and people started to run for safety.
According to The Times of India, Prime Minister Narendra Modi has announced 200,000 rupees (£2,460; $2,997) compensation for the families of the dead.
"I have spoken to officials & asked them to ensure all possible help to those affected due to the stampede in Varanasi," Mr Modi tweeted.
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