The statement said the referendum was contested on an unlevel playing field, and the two sides in the campaign did not have equal opportunities.
President Recep Tayyip Erdogan, however, claimed the OSCE's findings were politically motivated and that Turkey may hold a referendum to decide whether to continue its push for European Union membership.
Opposition parties complained of a number of irregularities in the voting, including an electoral board decision to accept as valid ballots that did not bear the official stamp.
There were protests in Istanbul with a few thousand people crowding the anti-Erdogan Besiktas and Kadikoy districts, blowing whistles and chanting "We are shoulder to shoulder against fascism".
"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".
Preliminary results showed 51.4 percent of voters had voted "yes".
"We have put up a fight against the powerful nations of the world", he told supporters greeting him at Ankara airport after arriving from Istanbul.
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"Know your place first", Erdogan told the monitors in an address to supporters outside his vast presidential palace in Ankara. "We'll continue on our path. Talk to the hand".
He added: "This country held the most democratic polls that have never been seen in any other country in the West". This risk will further be accentuated by worsening relations with Europe, Turkey's biggest trading partner, with which Mr Erdogan provoked conflicts in the lead up to the vote.
They also said the restrictions on media outlets and arrests of journalists ensured the yes campaign backed by President Recep Tayyip Erdoğan and the ruling party received the lion's share of coverage, tilting the odds in their favour. Monitors from the Organization for Security and Cooperation in Europe, who also listed numerous problems, said the move undermined important election safeguards.
Mr Trump has been criticised for congratulating Mr Erdogan, who is seen as taking Turkey in an increasingly authoritarian direction.
"The narrow result of the vote shows how deeply split the Turkish society is", Germany's Chancellor Angela Merkel and Foreign Minister Sigmar Gabriel said in a joint statement. "This suggests that Erdogan will become more robust and more challenging to deal with". The vote has fallen under suspicion because ballots that didn't carry an official stamp were allowed to be tallied. The main opposition party has demanded the result be nullified, saying the voting was marred by irregularities.
The victory of the yes campaign will consolidate Erdoğan's power, allowing him to run for two more election terms and potentially stay in power until 2029. The result triggered a 2 percent rally in the Turkish lira from its close last week. Germany, home to several million Turks, said it was up to Erdogan himself to heal the rifts that the vote had exposed.
The referendum approves 18 constitutional amendments to replace Turkey's parliamentary system with a presidential one.
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