Demonstrators had held a four-day sit-in at Gezi Park, angry at plans to redevelop that part of Taksim Square.
An Istanbul court later ordered the temporary suspension of the project to uproot trees in the park.
But there is wider anger against Prime Minister Recep Tayyip Erdogan and the unrest has spread to Ankara.
Protests have also been reported in other cities, including Bodrum, Konya and Izmir.
The US later expressed its concern over the reported number of injuries in Istanbul.
"We believe that Turkey's long-term stability, security and prosperity is best guaranteed by upholding the fundamental freedoms of expression, assembly and association, which is what it seems these individuals were doing,'' state department spokeswoman Jen Psaki said.
"These freedoms are crucial to any healthy democracy."
Earlier this month, riot police clashed with tens of thousands of people attempting to hold a May Day march in Istanbul.
By Alper BalliBBC Turkish
People are angry and frustrated at the government's decision to build a shopping centre at the expense of one of Istanbul's most famous parks. But what started as a protest against the uprooting of trees is turning into a wider platform for expressing anger against government policies.
Environmentalists have been joined by gay and lesbian groups, as well as socialists, union workers, members of opposition parties from across the political landscape and even so-called "anti-capitalist Muslims". The excessive use of force by the riot police and the insistence of the government to pursue their plans for the park have escalated tensions.
Taksim Square has political significance for the anti-government protests. As part of the reconstruction plans, the square was banned as a venue for this year's May Day rally and any kind of demonstrations in the future; but now the government's heavy handed approach risks turning the square into a focal point for protests against its policies.
What started out as an environmental protest in Istanbul became anti-government in tone, correspondents say.
The controversial redevelopment project is aimed at easing congestion around Taksim Square, but also involves building a shopping centre over Gezi Park.
Opponents of Mr Erdogan's plans say the park is one of the few green areas left in central Istanbul.
One banner at Friday's protest included a cartoon of Mr Erdogan dressed as an Ottoman sultan with the slogan: "The people will not bow down to you."
"We do not have a government, we have Tayyip Erdogan," political scientist and protester Koray Caliskan told the Reuters news agency.
"They are not listening to us," he added. "This is the beginning of a summer of discontent."
A dozen people have reportedly been admitted to hospital in Istanbul following Friday's clashes.
Hurriyet Daily News reported that seven of those wounded had serious injuries, including a broken leg and head injuries.
A journalist was hit in the head with a tear-gas canister and Hurriyet's own photographer was injured, it added.
There are also reports that foreigners were among those injured.
In all, more than 60 people were arrested.
In Ankara, protesters staged what they described as a solidarity rally, with many participants chanting: "Everywhere is resistance, everywhere is Taksim!"
Mr Erdogan has stressed that he is determined to go ahead with the redevelopment project in Istanbul.
Amnesty International condemned the police's tactics against "peaceful protesters".
"The Turkish authorities must order police to halt any excessive use of force and urgently investigate all reports of abuse," said John Dalhuisen, director of Amnesty's Europe and Central Asia Programme.
"They have a duty to ensure that people can exercise their right to free expression and assembly."
Mr Erdogan has been in power since 2002 and some in Turkey have complained that his government is becoming increasingly authoritarian.
Last week, Turkey's parliament approved legislation restricting the sale and consumption of alcoholic drinks.
The regulations would prohibit retail sales between 22:00 and 06:00, ban all alcohol advertising and promotion, and stop new shops and bars from opening within 100m (330ft) of schools and mosques.
Mr Erdogan said he wanted to stop young Turks from "wandering about in a state of inebriation" and was not trying to impose Islamic values.
The prime minister's Justice and Development (AK) Party has its roots in political Islam, but he says he is committed to Turkey's state secularism.
At least 12 people have been injured after Turkish police used tear gas and water cannon against protesters occupying a park in central Istanbul.