Incumbent Recep Tayyip Erdogan says that unofficial results from Turkey’s presidential election show that he won outright in the first round.
Erdogan also said the alliance led by his Justice and Development Party (AKP) has won a parliamentary majority.
However, opposition parties claimed that it was too early to announce the outcome of the June 24 presidential and parliamentary polls.
Under Turkey's new constitution, due to come into force after the elections, the president will hold expanded powers.
"The nation has entrusted to me the responsibility of the presidency and the executive duty," the 64-year-old said in televised remarks from Istanbul.
"The message is clear," he also said. "With turnout of nearly 90 per cent, Turkey has taught the whole world a democracy lesson."
Erdogan had 53.7 percent in the presidential poll with more than 87 percent of the votes counted, state media reported.
His closest rival, Muharrem Ince, stood at 30.2 percent.
With 88 percent of the votes for parliament counted, the AKP was leading with 43 percent, according to the Anadolu state news agency.
Ince's main opposition Republican People's Party (CHP), a secularist grouping, had 22 percent.
The pro-Kurdish Peoples' Democratic Party (HDP) looked set to cross the 10 percent threshold needed to win seats in parliament.
But the opposition disputed Anadolu's figures, saying many votes have yet to be counted, the daily newspaper Hurriyet reported.
It also alleged manipulation in the initial results being released by the state news agency.
Turnout was around 87 percent for both contests.
Amid signs of a weakening economy, Erdogan in April declared that presidential and parliamentary elections would be held on June 24, 17 months earlier than planned.
Under constitutional amendments approved after a controversial 2017 referendum, Turkey is making a transition from a parliamentary system to a presidential one -- giving the next president expanded powers, abolishing the prime minister's post, and eliminating many of the checks and balances designed to help parliament protect against the misuse of presidential powers.The changes will take effect after the elections.
"Turkey is staging a democratic revolution," Erdogan told reporters in the polling station in Istanbul where he voted earlier in the day. "With the presidential system, Turkey is seriously raising the bar, rising above the level of contemporary civilizations."
Polls suggested that the presidential vote could head into a second-round runoff on July 8 and that the Justice and Development (AK) Party could lose its parliamentary majority after 16 years.
Erdogan's leading challenger is likely to be Muharrem Ince of the main opposition Republican People's Party (CHP), a secularist grouping.
Ince, the CHP leader, is backed by the newly formed National Alliance, which along with secular social democrats, includes center-right conservatives, nationalist liberal conservatives, and conservative Islamists.
"I hope for the best for our nation," Ince said as he voted in his native port town of Yalova, south of Istanbul. Ince also vowed to spend the night at the headquarters of Turkey's election authority in Ankara to ensure a fair count.
Meanwhile, Turkey's election board said on June 24 that it was investigating complaints of alleged election safety breaches in the mainly Kurdish southeastern province of Sanliurfa.
Bulent Tezcan, a spokesman for Ince's CHP, said party observers had been kept away from polling stations in the southeastern Sanliurfa Province with "blows, threats, and attacks."
In Suruc, the scene of deadly clashes between CHP and AKP supporters 10 days before the vote, Tezcan said "armed people" were "running around openly and threatening the electoral atmosphere."
The CHP on June 24 formally complained, asking election officials to take action following reports that CHP observers were being blocked from monitoring the vote and had been physically assaulted.
"Necessary administrative and legal steps have been taken regarding claims on security issues at some polling stations in Suruc," election body chief Sadi Guven said.
Ince sent a message on Twitter in response to the alleged violence, urging his supporters not to lose hope in the face of the security issues and to cast their votes.
"Whatever they do, they are going to lose," Ince tweeted. "The era of winning elections with fraud is over. I will protect your votes at the risk of my life, and we will succeed."
National Alliance parliamentary candidates vowed that if they secure majority control of the legislature, they will try to roll back the Erdogan-backed constitutional amendments narrowly approved in the controversial 2017 referendum.
On the eve of the elections, hundreds of thousands of people packed the Maltepe shore on the Asian side of Istanbul for Ince’s final speech, which followed massive rallies in Izmir, Ankara, and elsewhere.
The former physics teacher, who has pledged to end what he calls Erdogan’s "one-man rule," claimed the gathering attracted 5 million people, although no official estimates were available.
Ince told the crowd that if he wins, “it will not just be Ince who wins...80 million people will win! Turkey will win!"
Erdogan, who held five rallies in Istanbul on June 23, called on supporters to be vigilant at the polling stations.
"Beware: do not be overcome by languor. Take all your friends, relatives to the ballot box," he said. "The security personnel will intervene once you ask them to do so."
Erdogan has dominated Turkish politics for more than 15 years, first as prime minister and since 2014 as president. He remains popular with vast segments of the population.
Selahattin Demirtas, a jailed Turkish-Kurdish politician, is also a candidate for the pro-Kurdish Peoples Democratic Party's (HDP). He has campaigned from jail, where he has been held since November 2016 while awaiting trial on charges of having ties to Kurdish militants. He denies the charges.
Kurds make up about 18 percent of Turkey's 78 million people, mostly in eastern regions, and are expected to play a key role in deciding the outcome of the parliamentary elections.
Nearly 60 million Turks, including more than 3 million expatriates, were eligible to vote in the elections for the presidency and for 600 parliamentary seats.