The deputy chairman of the People's Republican Party (CHP) said there had been complaints in many regions that people had been blocked from voting in private - and that some ballots had been counted in secret.
"The only decision that will end legitimacy debate and ease people's concerns about the judiciary is for the High Election Board to cancel the referendum", Tezcan said.
He said it was not possible for authorities to determine how many ballot papers may have been irregularly cast.
During the campaign, Mr Erdogan repeatedly criticised European countries, including Germany and the Netherlands, accusing them of "Nazi-like" tactics for banning his ministers from speaking to rallies of Turkish voters overseas. The OSCE also said the vote did not meet standards set by the Council of Europe.
Turkish President Recep Tayyip Erdogan has finally fulfilled his long-held ambition to expand his powers after Sunday's referendum handed him the reigns of his country's governance.
Despite a state of emergency and a widespread crackdown on dissent, Erdogan succeeded in persuading only 51.4% of voters to back his constitutional upheaval.
While the final results of the referendum will be declared in 11-12 days, the electoral board has already confirmed victory for the "Yes" side.
Erdogan reaffirmed he would now hold talks on reinstating capital punishment, a move that would automatically end Turkey's European Union bid, and would hold a referendum if it did not get enough votes in parliament to become law.
The meager 51.4 percent "yes" vote shows a divided Turkey. Putin came to power in 1999 and has served two nonconsecutive terms as president since then. Since now, the prime ministerial power planned to be abolished and replaced with a presidential scheme.
OSCE argued that under the state of emergency put in place after the July 2016 failed coup attempt, fundamental freedoms essential to a genuinely democratic process were curtailed. The system is created to ensure that only one vote is cast per registered person and to avoid the possibility of ballot box-stuffing.
She said that the president and ruling AKP were given preferential treatment in the media and more air time, and that the jailing of journalists and closure of media organizations after the attempted coup past year had "led to widespread self-censorship in Turkey". "They shouldn't try, it will be in vain", he told cheering, flag-waving supporters in Istanbul.
But there were just as many who were devastated by the result. The EU will likely halt negotiations for Turkish accession, although they likely won't be formally suspended, Sinan Ülgen of Carnegie Europe said. "It's too late now". Wasin Yalcin, 24, said the vote represented "a new hope for us to get rid of foreign forces", while Yusuf Basaran, 20, said he believed "Europe's spine has cracked".
He also took a dig at global critics.
March 2014 - After Erdogan threatens to "eradicate" Twitter at a campaign rally, Turkey bans the social media site, and a two-week countrywide blackout ensues. Under the new "presidential system" he will be both head of state and head of government for the first time.
Considering how tilted critics called the campaign, NPR's Peter Kenyon in Istanbul says many were surprised at how close the vote was.
Turkey's President Recep Tayyip Erdogan waves to supporters in Istanbul, Turkey, on Sunday, April 16, 2017.
He said: "The result will depend on how far the opposition will take their claim of irregularity in the voting, and what the global reaction will be".
"Russia should be very cautious about Erdogan and his actions".
The referendum campaign was highly divisive and heavily one-sided, with the "yes" side dominating the airwaves and billboards. The "no" campaign complained of intimidation, detentions and beatings.
German integration commissioner Aydan Ozoguz warned against criticizing Turks living in Germany across the board over how they voted, telling regional newspaper Saarbruecker Zeitung that only around 14 percent of all German Turks living in Germany had voted "Yes" and added that most migrants had not voted.
With his new authority, Erodgan will be able to appoint Cabinet ministers, issue decrees, choose senior judges and dissolve parliament, the Grand National Assembly.
Opposition parties still cried foul.