BOGOTÁ, Colombia—Protests against President Iván Duque stretched into a third-straight day with marchers accusing security forces of brutality and government officials claiming that an outbreak of looting and vandalism was part of an orchestrated terror campaign.

The demonstrations in Colombia are the latest to roil South America and have Mr. Duque, a 43-year-old conservative U.S. ally, facing his worst crisis since he took office 15 months ago.

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