At least eight people were injured, hospital officials said.
Turkish warplanes flew over the demonstrators in Diyarbakir, the largest city in overwhelmingly Kurdish southeastern Turkey. Many of the demonstrators shouted support for autonomy-seeking guerrillas.
Several police officers were among the eight injured during the clashes, hospital officials said. Television footage showed police trying to defend themselves with shields against a shower of rocks.
Demonstrators shouted slogans in praise of imprisoned Kurdish rebel leader Abdullah Ocalan and unfurled giant pictures of Ocalan and banners of his guerrilla group, the Kurdistan Workers Party, or PKK, television footage showed.
The PKK is listed as a terrorist organisation by the US and the European Union.
The spring festival of Nowruz has been the scene of clashes in the past, especially in the early 1990s, at the height of a conflict between the Turkish army and Kurdish rebels.
The prime minister, Recep Tayyip Erdogan, warned on Tuesday against provocations during Nowruz celebrations and urged all to hold festivities in peace.
Jumping over a fire in Diyarbakir
during Nowruz celebrations
The family of an 18-year-old Kurdish youth said the teen was shot and wounded by police during an illegal demonstration in Istanbul on Monday evening, private NTV television reported. There was no immediate comment from police.
Kurds have been celebrating Nowruz since Sunday, singing, dancing and jumping over fires, symbolically burning away past impurities. Nowruz, the Farsi word for "new year", is an ancient Persian festival, celebrated on the first day of spring in several countries, including Afghanistan and Iran.
The festival is mainly marked by Kurds in Turkey and has traditionally been used by Kurds to express their support for Kurdish fighters who launched a war for autonomy in 1984. The fighting has claimed about 37,000 lives.