The local fire service said three casualties had been rescued from the rubble.
The Foreign Office has warned of possible aftershocks and advised Britons in surrounding areas to follow the advice of local authorities and tour operators.
A number of holidaymakers, some hundreds of miles from the epicentre, have spoken of their fear when they felt tremors as the quake struck.
Former footballer Kevin McNaughton said he is near Dalaman, around 120 miles from Bodrum.
The former Cardiff City player tweeted: "Jesus just experienced earthquake in Turkey, literally crapped myself room shaking allover place, Just stood outside now no sure what to do."
Many people ran from their homes or holiday apartments with pillows and blankets, according to one Briton who is staying in Didim, around 60 miles from Bodrum.
Kristian Stevens, from Nelson in Lancashire said he felt the building he was in "shake like a jelly".
The 48-year-old told the Press Association he had just gone to bed when the quake struck at around 1.30am local time.
He said: "It was quite surreal as I had just laid down in bed and the whole building shook. The whole building shook like a jelly.
"Many of the locals rushed out into the streets still in underwear.
Earthquake rocks Turkey-Greek coasts
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"Some have been seen with blankets and pillows not sure if it is safe to return home."
Sophie Wild said she ran from her third floor accommodation when she woke to a loud banging noise.
The 21-year-old from Canterbury in Kent is coming to the end of her holiday in Altinkum, around 500 miles away from Bodrum.
She said: "We were asleep and were awoken by what sounded like banging on our door, it got louder and louder and the building started shaking. We jumped up ran to the balcony to see what it was (my first thought when we heard the banging was that we were being attacked).
"When we realised it was an earthquake, we got an immediate sense to get out, we thought the building was going to crumble around us. We ran down our stairs (were on third floor).
"People were running out of rooms, banging on people's doors to make sure they were out. Everyone just ran outside and waited for a couple of hours - it's only now that people are starting to go back to their rooms. There are a few cracks in the walls but otherwise staff says it's safe."
Daniel Markham, a councillor on Tonbridge and Malling Borough Council in Kent, said he too felt the effects on the Greek island of Rhodes.
He tweeted: "£earthquake £Rhodes felt it here too. Pretty strong. Looked out the window to see the waves in the pool."
A spokesman for the Foreign Office said: "We are speaking to the Turkish and Greek authorities following an earthquake off the coast of Bodrum and near the island of Kos.
"Any British people in the areas affected should follow the instructions of local authorities."
Press Association
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