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The EU and rights groups have accused Ankara of using its broad anti-terror legislation to intimidate journalists and stifle dissent.
Ankara rejects this, saying it needs the laws to fight militant groups.
The visa-free deal is supposed to be in place by the end of June, but that timetable looks increasingly unlikely, the BBC's Jonathan Blake in Strasbourg reports.
The European Commission earlier this month said it was satisfied that the majority of the 72 conditions had been fulfilled by Turkey.
But the European Parliament is refusing to vote until all the criteria are met, our correspondent says.
The deal was offered in return for Turkey taking back migrants who crossed the Aegean Sea to Greece.
The EU fears that without it, Turkey will not control migration.
Five benchmarks still to be met by Turkey:
Corruption: Turkey must pass measures to prevent corruption, in line with EU recommendations
Data protection: It must align national legislation on personal data protection with EU standards
Europol: An agreement is to be concluded with the continent's law enforcement agency
Judicial cooperation: It must work with all EU members on criminal matters
Legislation on terrorism: Turkey is also required to bring its terror laws in line with European standards
Turkey has threatened to stop taking back migrants from Greece if the EU fails to deliver on visa liberalisation.
The large influx of migrants and refugees arriving in Europe from Turkey, and from North Africa, has caused a political crisis among EU states.
Under the EU-Turkey agreement, migrants who have arrived illegally in Greece since 20 March are to be sent back to Turkey if they do not apply for asylum or if their claim is rejected.
For each Syrian migrant returned to Turkey, the EU is to take in another Syrian who has made a legitimate request.
Turkey in numbers
2.75m Syrian refugees registered with UN
151 out of 180 countries on World Press Freedom Index
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.