Six people were injured in the blast at the office of the People's Democratic Party (HDP) in the southern city of Adana caused by a suspect parcel, three of them seriously, a party official told AFP.
A bouquet of flowers sent to the party's office in the nearby city of Mersin also exploded, the official said. Video footage showed several people with bloodied faces.
The HDP's co-chairman Selahattin Demirtas was due to address a rally in Mersin this evening and the party said there was no suggestion of him cancelling the event.
"I strongly condemn this attack," Davutoglu told a rally in the city of Karaman in central Turkey.
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Davutoglu said he gave a "clear instruction" for a full-scale investigation. But he warned against any smear campaign to discredit his ruling party after some HDP figures blamed the government for the attacks.
"We have stood against violence since the very beginning. God willing, we will march into June 7 in peace," he said.
But in a public rally in the Black Sea city of Samsun, President Recep Tayyip Erdogan lambasted the HDP for its links to the outlawed Kurdistan Workers' Party (PKK) which has waged an armed insurgency in the southeast for Kurdish autonomy.
"I am appealing to all of Turkey: Turks, Kurds, Arabs, Circassians, Abkhazians - whoever comes to mind - will 78 million of you give the right response to the political organisation that is guided by a terrorist group?" he said.
His spokesman Ibrahim Kalin strongly denounced the explosions and dismissed accusations targeting the government as "unacceptable".