Three soldiers were also killed today in a roadside bombing by the outlawed Kurdistan Workers' Party (PKK) in the town of Cizre in the Kurdish majority southeast, the army said.
The baby, named Miray, was hit in the head when the family's house in Cizre was shelled on Friday night during street battles between Turkish forces and PKK rebels.
Her 73-year-old grandfather Ramazan Ince was then shot while he was carrying Miray to the ambulance, witnesses told AFP.
The family claims that the fire came from the direction of the state hospital in Cizre, which is controlled by Turkish security forces.
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Tensions are running high throughout Turkey's restive southeast as security forces impose curfews in several towns including Cizre in a bid to root out PKK rebels from urban centres.
A bomb planted by "members of the separatist terrorist organisation" went off today as an armoured vehicle passed by in Cizre, killing three soldiers, the army said in a statement, referring to the PKK.
The military offensive marks a new escalation in five months of fighting since the collapse of a ceasefire between the government and the rebels in July.
Residents of the towns under curfew are facing food shortages and problems with water and electricity supplies. Many homes have been damaged by shelling.
The army said yesterday that almost 200 militants have been killed in the latest military operation against the PKK since mid-December.
A cultural centre in the city of Sirnak was attacked Friday by PKK rebels with Molotov cocktails, and four students were injured, the state-run Anatolia news agency reported.