Ex-BBC journalist Jacky Sutton, 50, was found dead in a toilet at the city's Ataturk airport between 17-18 October.
She was the acting Iraq director for the Institute for War and Peace Reporting (IWPR) and was travelling from London to Irbil, northern Iraq.
In a statement Jenny Sutton, Ms Sutton's sister, said "at this stage we believe that Jacky acted alone".
But the statement added that there remained "the possibility that additional information will come to light".
An investigation into the death is currently being carried out by Turkish authorities.
The statement said the family and the IWPR had carried out an "extensive review" of information provided by Turkish authorities.
Jenny Sutton said: "The family is satisfied with the investigation undertaken by the Turkish authorities.
"We were deeply sceptical about initial reports. But based on the evidence we have seen, at this stage we believe that Jacky acted alone."
The family and the IWPR said the Turkish authorities had provided access to "the complete dossier of evidence" relating to the case, including copies of CCTV images from the airport and witness statements.
This all indicated that Ms Sutton had been alone and there was no evidence of a struggle, they said.
The decision to release information from the investigation had followed "widespread speculation" about the circumstances surrounding Ms Sutton's death, the statement added.
The IWPR and the family said they would seek a further assessment from an independent investigative expert to confirm the findings once the Turkish investigation was complete.
The London-based IWPR supports local journalism in countries affected by conflict and crisis.
Its previous Iraq director, Ammar Al Shahbander, was killed in a car bomb attack in Baghdad on 2 May and a memorial service had been held for him in London last week, according to the IWPR website. Ms Sutton had been in London attending the service.
Ms Sutton worked for the BBC World Service between 1998 and 2000, reporting from Africa, the Middle East and London.
She also served in senior roles at the United Nations.
The IWPR described Ms Sutton as "extremely bright" and "universally loved".
No other parties were involved in the death of a British woman at an Istanbul airport, her family believes.