Istanbul (AFP) - Thousands of pro-Kurdish demonstrators incensed by Turkey's inaction in the fight against jihadists on the Syrian border clashed with police across the country on Tuesday, leaving at least a dozen dead and many wounded.

Five people were killed in Diyarbakir, the largest town in Turkey's majority-Kurdish southeast region, according to press reports.

Several other deaths were recorded in other southeastern towns, including three killed in Mardin, two in Siirt, one in Batman and another in Mus, while police used tear gas and water cannon to disperse angry protests in Istanbul and Ankara.

Thousands of people had joined the demonstrations called by the main pro-Kurdish party, the People's Democratic Party (HDP), against Ankara's failure so far to intervene militarily against Islamic State jihadists fighting for the Syrian border town of Kobane.

Prime Minister Ahmet Davutoglu has vowed that Turkey will do whatever necessary to prevent the fall of Kobane, also known as Ain al-Arab.

But Kurds bitterly accuse Ankara of merely looking on as the town risks being overrun by jihadists despite dozens of Turkish tanks being deployed on the border.

- 'Violence will be met with violence' -

Turkish Interior Minister Efkan Ala accused the pro-Kurdish protesters of "betraying their own country" and warned them to disperse or face "unpredictable" consequences.

"Violence will be met with violence... This irrational attitude should immediately be abandoned and (the protesters) should withdraw from the streets," Ala told reporters in Ankara.

In Mus, a 25-year-old protester was killed after being struck in the head by a tear gas cannister fired by police to disperse the protesters.

In Diyarbakir, five were killed by gunshots in clashes between pro-Kurdish activists and Islamists.

Enraged youths in the southeastern town had overnight torched a police vehicle, scores of other vehicles and shops and attacked government offices.

In Istanbul's Gazi neighbourhood, largely populated by Kurds, police used tear gas and water cannon to disperse a protest by several hundred Kurds, an AFP correspondent said.

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Elsewhere in Istanbul, one person was seriously injured after being shot in the head from close range.

- Nationwide protests -

Local authorities ordered a curfew in several Kurdish-majority provinces including Diyarbakir, Mardin, Siirt and Van.

Kurds have been particularly irked by the reluctance of Turkish authorities, who are concerned by Kurdish separatism, to allow Kurds over the border to fight Islamic State.

They have warned that the fall of Kobane could mean an end to the peace talks between Ankara and the Kurdistan Workers' Party (PKK), which waged a deadly insurgency in Turkey for the last three decades but has largely observed a ceasefire since last year.

Jailed PKK leader Abdullah Ocalan said in a message relayed by his brother that the government had until mid-October to show it was serious about the peace process.

"They (the government) are talking about resolution and negotiation but there is no such thing," he said.

"This is an artificial situation, we will not be able to continue anymore," said the statement carried by the Firat news agency.

"The state must take action... Can a peace process make any progress this way?"

Kurdistan Communities Union (KCK) -- considered the urban wing of the mountain-based PKK -- called on "millions" to take to the street to protest against what it termed "IS brutality".