The legislation passed late Wednesday expands the remit of Turkey's audiovisual authority RTUK, which regulates and supervises radio, television and on-demand broadcasts.
Online platforms must now apply to RTUK for licences to broadcast and the authority can give fines as well as block access to broadcasters.
Opponents say the law will mean that internet giants like Netflix and YouTube could be blocked in Turkey if complaints are made against some of their videos.
Garo Paylan, an MP of the pro-Kurdish Peoples' Democratic Party, said that if a platform like Netflix published a video in which there was, for example, criticism of President Recep Tayyip Erdogan, they risked being blocked if the content was not removed.
"This can make Turkey become a completely closed country. We could fall into the same league as North Korea. We could become a country where Netflix, Wikipedia and YouTube are blocked," Paylan told AFP.
Wikipedia has been blocked in Turkey since April 2017 because Ankara says it failed to remove content that promoted terror and accused Turkey of cooperation with terror groups.
There have been blocks on YouTube in the past, but it is currently accessible in Turkey.
Opponents worried the government was tightening controls ahead of elections in 2019.
"As the local and general elections near, the government is tightening the screws," Baris Yarkadas of the opposition Republican People's Party (CHP) said on Twitter.
"The new regulation against websites is the most vile censorship," he added.
The RTUK watchdog says it is an autonomous regulator, with its nine members elected by lawmakers on the basis of the number of MPs of each political party.
A majority of lawmakers in Turkey's parliament are from Erdogan's ruling Justice and Development Party.
When the law was first presented in February, Transport and Communication Minister Ahmet Arslan denied there was any censorship.
He said the authorities must take measures against TV and radio shows if there are "wrongs committed against national security or the country's ethical values".
But Paylan said the opposition was able to force a significant amendment through to ensure private individuals who post video online would not be affected by the new law.
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