This is the country's worst unrest in a decade, and more is expected ahead of the election early next year.
The Uganda police pathologist and the head of police health services told journalists in the capital, Kampala, that they had counted 37 bodies by Thursday morning.
The protests broke out on Wednesday after police arrested Wine, who was expected to appear in court on Thursday in the eastern town of Iganga.
Police accuse him of flouting COVID-19 guidelines that require presidential candidates to address less than 200 people.
Police have said 350 people were arrested and detained throughout Kampala.
Tension remained in the capital Thursday, with a heavy military and police presence.
Wine, who has been arrested many times in recent years, has captured the imagination of many Ugandans with his persistent calls for President Yoweri Museveni to retire after 36 years in power.
Speaking to reporters, Security Minister Gen. Elly Tumwine warned the protesters that they will be dealt with if they continue.
”This was a deliberate and pre-planned move to cause chaos because we have evidence," Tumwine asserted.
“But I want to warn those inciting violence that they will reap what they sow.”