Turkey's Kurdish minority -- estimated to make-up around a fifth of the population -- is seen as a crucial constituency in the poll and winning the support of some Kurds could help the 'Yes' vote over the finish line.
Many Kurds back the pro-Kurdish Peoples Democratic Party (HDP) but others, especially from more conservative families, have sympathy for Erdogan's Islamic-rooted Justice and Development Party (AKP).
A crucial issue for all Kurds, however, is whether after the referendum there will be a resumption in the peace process to end the over three-decade insurgency by the Kurdistan Workers Party (PKK) after a ceasefire collapsed in 2015.
"I want a constitution under which everyone can be represented equally," Hikmet Aydogan told AFP as he voted in the main Kurdish city of Diyarbakir, where support for the HDP has traditionally been strong.
"I want the Kurds to succeed. The Kurds must emerge successful out of the vote."
Ahmet Kemal Cengiz, 39, a farmer, said: "I don't want to say what I voted for but whatever happens, I hope will be good for our country."
He added: "Let whatever to emerge from the ballot box be good for mankind and for the Kurds."
Voting was marred by shooting between two groups in the garden of a school in the Yabanardi village in Diyarbakir's province.
Two people were killed and three others were wounded in the fighting whose cause was not immediately clear although some alleged it was triggered by alleged irregularities in the voting.
The referendum campaign was dominated in its last days by debates over a possible federal system in Turkey after Erdogan's advisor Sukru Karatepe suggested, if approved, the constitutional changes could lead to federalism.
This could imply the creation of a Kurdish region in southeastern Turkey, something yearned for by many Kurds.
But Erdogan himself denied the comments in a public rally this week after an angry backlash from his major ally the right-wing Nationalist Movement Party (MHP), saying that any federal system was off the table.
'Path of peace'
The HDP, Turkey's second-largest opposition party, has called for a 'No' vote but complained of a media embargo that allowed its MPs just minutes of airtime on TV during the campaign.
The HDP's co-leaders Selahattin Demirtas and Figen Yuksekdag are currently in prison as well as 11 other HDP MPs accused of links to the PKK, in what the party says was a deliberate move to eliminate them from the campaign,
A picture of Demirtas's wife Basak casting her vote in Diyarbakir was shared widely on Twitter by supporters.
"I hope that the result of this referendum will help our people move on the path of peace, democracy and freedom," she said.
HDP MP Feleknas Uca, who voted in Diyarbakir, said April 16 was only a "beginning" for the future of all communities including the Kurds.
"Today is a beginning for the future of all peoples including Kurds, Turks, Arabs, Armenians, Assyrians and Roma," she told reporters.
"The 'No' vote will be the voice of freedom and peace. The future of the people lies in No," she said.
"It's the people's voice which will pour water into the fire. It's the people's voice which will bring a solution."
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