Demonstrators in Istanbul's central Besiktas district chanted slogans, calling Erdogan a "murderer" and a "thief."
The protestors accuse the election commission, which earlier rejected all opposition requests to annul the referendum, of partisanship.
Protests were also planned for other areas of Istanbul and in other Turkish cities.
Police had arrested 38 in Istanbul in raids early Wednesday, following protests the previous night, according to reports by opposition newspaper Birgun.
Those arrested include Mesut Gecgel, the Istanbul Province chairman of the left-wing Freedom and Solidarity Party (ODP), who helped to organize the demonstrations in Besiktas, Gecgel wrote on Twitter.
President Erdogan has claimed victory in the race on empowering his office, with state media saying the "yes" camp garnered 51.4 per cent of the vote.
However, in Turkey's three largest cities - Istanbul, Ankara and Izmir - a majority voted "no."
Opponents of the reforms, which grant sweeping powers to the presidency, claim there were irregularities and possible fraud on election day, pointing in particular to the election commission's decision allowing unsealed ballots to be counted.
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