The death toll, which currently stands at more than 3,400 people, has increased rapidly since the first earthquake struck early on Monday morning.
About 12 hours later, a second powerful tremor hit further north.
Rescuers have been combing through mountains of rubble in freezing and snowy conditions to find survivors.
Countries around the world are sending support to help the rescue efforts, including specialist teams, sniffer dogs and equipment.
The US Geological Survey said the 7.8 magnitude tremor struck at 04:17 local time (01:17 GMT) at a depth of 17.9km (11 miles) near the city of Gaziantep.
Seismologists said the first quake was one of the largest ever recorded in Turkey. Survivors said it took two minutes for the shaking to stop.
The second quake – triggered by the first – had a magnitude of 7.5, and its epicentre was in the Elbistan district of Kahramanmaras province.
Many aftershocks are still being felt across the region.
The number of deaths are likely to increase as much as eight times, as rescuers find more victims in the rubble.
Many of the victims are in war-torn northern Syria, where millions of refugees live in camps on both sides of the border with Turkey. There have been dozens of fatalities reported in rebel-held areas.
Thousands of buildings across both the countries have collapsed, and several videos show the moment they fell, as onlookers ran for cover. Many buildings that were as large as 12 storeys high are now flattened, roads have been destroyed and there are huge mountains of rubble as far as the eye can see.
Among the buildings destroyed was Gaziantep Castle, an historic landmark that has stood for more than 2,000 years.
Turkey’s energy infrastructure has also been damaged, and videos have emerged showing large fires in southern Turkey. Social media users claimed they were caused by damage to gas pipelines.
Turkey’s energy minister Fatih Donmez confirmed there had been serious damage to the infrastructure, but did not mention the explosions.
Turkey lies in one of the world’s most active earthquake zones.
The Turkish Red Crescent has called for citizens to make blood donations, and the organisation’s president, Kerem Kınık, said on Twitter that additional blood and medical products were being sent to the affected region.
Following an international appeal for help, Turkey’s President Erdogan said 45 countries had offered support.
UN Secretary General António Guterres has called for an international response to the crisis, saying that many of the families hit by the disaster were “already in dire need of humanitarian aid in areas where access is a challenge”.
The European Union is sending search and rescue teams to Turkey, while rescuers from the Netherlands and Romania are already on their way. The UK has said it will send 76 specialists, equipment and rescue dogs.
France, Germany, Israel, and the US have also pledged to help. Russian President Vladimir Putin has offered help to both Turkey and Syria, as has Iran.
Turkey’s interior minister, Suleymon Soylu, said 10 cities were affected by the initial quake including Hatay, Osmaniye, Adiyaman, Malatya, Sanliurfa, Adana, Diyarbakir and Kilis.
School has been suspended in those cities for at least a week.