The activists were marking the first anniversary of the death of Berkin Elvan who died on March 11, 2014, after spending 269 days in a coma due to injuries sustained when he was hit by a tear gas canister fired by police in the mass protests of early summer 2013.
He was aged 15 at the time of his death.
Elvan's death in 2014 drew huge protests in Istanbul and across the country but it appeared the rallies marking the anniversary would not be on this scale.
The protesters refused to budge as the plain clothes officers arrested them, holding tightly to each other, before being marched away by police. Ten people were arrested, the CNN-Turk channel reported.
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In the capital Ankara, police used water cannon to break up a protest in the city, arresting 11 people, local media said. Police were also reported to have broken up a demonstration in the western city of Izmir.
More protests were expected in Istanbul and other Turkish cities later in the day.
The demonstrations began as an action against the redevelopment of Gezi Park adjoining Taksim Square in Istanbul but snowballed into a full scale nationwide wave of protest against Erdogan, who was then premier.
Elvan's parents and supporters are furious that no police officer has been brought to trial over his death, accusing the authorities of concealing the name of the perpetrator.
They say he was not even actively protesting but was caught in the turmoil in Istanbul after going out to buy bread.
Erdogan had sparked an outcry after he called Elvan a thug with links to a "terrorist organisation" and encouraged his supporters at a rally to boo Elvan's mother.