Police are now convinced a network of suspects either from Turkey or using fake passports from there are responsible for the August 17 attack.
They are also investigating a possible link with the political grievances of Uighurs, a Turkic, mostly Muslim people from China's western region of Xinjiang.
However Thai authorities have issued a directive for officials not to refer to Uighurs when discussing the bombing and to call the attack a "disturbance", not terrorism.
Thailand's military government is worried talk of terrorism will damage the country's tourism industry which accounts for 10 per cent of the nation's faltering economy.
The military seized power last year promising it would keep Thailand safe after years of political turmoil.
Thailand infuriated the Uighur movement in July when the country deported 109 Uighurs to China where human rights groups said they would face harsh treatment from Chinese security agencies.
Most were men separated from their wives and children.
Uighurs claim they have long been persecuted by Chinese authorities and an estimated 3000 of them are living in Thailand.
Thai investigators have for the first time named a Turkish national who is wanted for possessing "war materials."
They said Emrah Davutoglu was the husband of Thai woman Wanna Suansan who is alleged to have rented rooms in a seedy Bangkok apartment building where bombs were assembled.
The couple are believed to be in Turkey.
"He is Turkish," said national police spokesman Prawut Thavornsiri, referring to Mr Davutoglu.
Investigators claim a series of breakthroughs in investigations in the hunt for those responsible for the unprecedented bombing of the Erawan shrine that killed 20 people and injured 120 others.
They say a man identified only as a foreigner but who speaks Turkish who was arrested on Tuesday near the border with Cambodia had admitted being near the shrine around the time of the attack but denied placing the bomb.
They said the man's fingerprints matched those found on bomb-making material discovered in an apartment raided last weekend.
Thai media have published a picture of a Chinese passport they say was found on the man.
The name on the passport is Yusufu Mieraili and his birthplace is given as Xinjiang.
Police said they did not yet know if it was a fake document.
Police suspect that man arrested on August 29 in a Bangkok room with bomb-making materials is also Turkish.
He has identified himself as Bilal Mohammed.
The bombing has prompted authorities in Bangkok to move to tighten Thailand's borders, including the introduction of a biometric system to crack down on fake travel documents and human trafficking.
The Bangkok Post reports that Thai authorities are also closely monitoring the movements of ethnic Uighurs in Thailand, looking for acts such as passport forgery.
Many Uighurs who have reached Thailand from China have pleaded with authorities to be allowed to travel to Turkey.
Good work Bangkok rozzers. Perhaps another 3 million baht reward is in order.
Posted by Arun Muruga on September 3, 2015 17:33