Her distraught father, mother, stepmother, brother and grandparents have all spoken of their pain following the grim discovery made by detectives hunting for the 16-year-old.
They spoke out as Avon and Somerset Police were granted an extra 24 hours to question a 28-year-old man and a 21-year-old woman on suspicion of murdering Becky.
Five other people – three men and a woman – also remain in custody after being arrested on suspicion of assisting an offender.
Becky’s father and stepmother Darren and Anjie Galsworthy said: “To receive such news as we did today about our dear daughter and stepdaughter Becky is too much to bear.
“We have the support of a good family network to help us in these challenging times.”
The teenager’s mother Tania Watts, brother Daniel and grandmother Pat said: “We are devastated to be told of this latest development.
“We are at an utter loss to understand why anyone would want to hurt our beautiful Becky in such a brutal way.”
John Galsworthy, speaking on behalf of the Galsworthy family, paid tribute to Becky as a “wonderful” daughter, sister, granddaughter and niece.
“We are heartbroken at what has been discovered overnight,” Mr Galsworthy said. “We cannot begin to understand what has happened and why.
“Becky was a wonderful daughter, sister, granddaughter and niece and we cannot put into words how devastated we are.”
Becky vanished from her home in Crown Hill, St George, Bristol at around 11:15am on 19 February.
She left home with her iPhone, laptop and tablet computer but did not tell friends or family where she was going, or take extra clothing.
She was reported missing at 4pm on 20 February and a large search operation – involving six police teams each day – was launched.
Detective Superintendent Mike Courtiour said information received by police late on Monday led them to a house at Barton Court in the Barton Hill area of Bristol where body parts were found.
“The information suggested that Becky’s body had been cut up and a search at the new location resulted in the discovery of body parts,” he added.
“Although formal identification procedures have not yet been completed, we have made Becky’s family aware.”
As part of the investigation, four men, aged 29, 29, 23 and 23, have been arrested on suspicion of assisting an offender, along with a 23-year-old woman.
Floral tributes have begun to be left close to the house at Barton Court where police made the grim discovery.
One simply read: “RIP Becky Watts.”