The masked protester scaled a flagpole at a military base in the Kurd-dominant Diyarbakir province on Sunday and removed the flag.
Protests have flared over the planned construction of a new army base in the province, a move that some Kurds view as a threat to the peace process.
On Saturday, two Kurds were fatally shot while clashing with government troops.
The prime minister vowed that the young protester who removed the national flag "will pay the price the same way as those who sent him there," and ordered the interior ministry to conduct an investigation.
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