Recep Tayyip Erdogan says he has been returned as president of Turkey, according to unofficial results.
But the main opposition, the Republican People’s Party, says Mr Erdogan could still fall short of the 50% needed to avoid a presidential runoff on 8 July.
People have been voting in both presidential and parliamentary elections.
The ballots will transform the country from a parliamentary democracy into an executive presidency – something approved by a narrow majority in a referendum last year.
Under the new system, the president will be able to form the government, appoint ministers, vice presidents and top bureaucrats, issue decrees, prepare the budget and impose states of emergency.
Opposition politicians had vowed to return Turkey to parliamentary democracy with strong checks and balances.
“The nation has entrusted to me the responsibility of the presidency and the executive duty,” 64-year-old Mr Erdogan said.
“I hope nobody will try to cast a shadow on the results and harm democracy in order to hide their own failure.”
Mr Erdogan, who has been in power since 2003, also declared victory for the People’s Alliance, which comprises his AK Party and the smaller Nationalist Movement Party.
He said the alliance had a “parliamentary majority” in a new 600-member assembly, containing 50 more members than before.
Deputy Prime Minister Mehmet Simsek said the apparent victory “sets the stage for speeding up reforms”.
Earlier, Mr Erdogan’s main rival for the presidency, 54-year-old former physics teacher Muharrem Ince, accused the country’s state-run Anadolu news agency of “manipulation”.
After the agency said 85% of ballot boxes had been counted, Mr Ince claimed it was only 37%.
Election observers and party members should not “abandon the ballot boxes”, he said.
But government spokesman Bekir Bozdag alleged Ince was “threatening and targeting” the agency.
Mr Erdogan said there had been no serious voting violations.
Mr Ince, backed by the centre-left Republican People’s Party, drew huge numbers at rallies in Istanbul, Ankara and Izmir.
Five candidates ran against Mr Erdogan in the presidential race – with one of them campaigning from a prison cell.