Police officers forced their way into an address in Allan Bedford Crescent, Costessey, at around 7am on Friday following a call from a member of the public.
The bodies of a 45-year-old man, named locally as Bartlomiej Kuczynski, a woman aged 36 and “two young girls” of school age were found.
Norfolk Police said all four were discovered with injuries and post-mortem examinations will be held in due course to establish the cause of death.
Three members of the family lived at the address permanently, while a fourth person was there “non-permanently”, the force said.
“The man and two children lived at the address, but the woman involved was visiting and didn’t live at the property. Next of kin have been informed and we are supporting them,” police said.
Officers were called to the same house on 14 December last year as part of a missing persons inquiry and they will now look back on that incident. Police are expected to remain at the scene for a “considerable time”.
At a press conference on Friday afternoon, officers said formal identification is yet to take place.
Detective Chief Inspector Chris Burgess from the Norfolk and Suffolk Major Investigation Team, said a weapon had been recovered from nearby woods but was not believed to be linked to the deaths at this stage.
“I would stress to the community, let’s be very aware of what we share on social media without confirmed information,” he added.
“We don’t want to cause any undue distress or give false information that might actually hamper the investigation rather than support it.”
A police cordon is still in place and detectives from the Norfolk and Suffolk Major Investigation Team are leading the enquiries while forensic examinations are carried out.
A neighbour told the BBC that she knew the family in passing as their children went to school together.
“I only met them sometimes in the street because my son and her kids go to the same school, so we met sometimes in school or in the street and said ‘hello’.”
Police believe it is an “isolated” incident and Detective Burgess said: “Our investigation is in the very early stages and this is clearly a very distressing and tragic incident.
“While we’re following a number of lines of enquiry, at this time we believe this is an isolated incident.
“I understand that news of this incident will be shocking to the local community and as soon as we can we will release more information.”
Daniel Sealey, who lives hundreds of yards from the house on the same estate, told the BBC he was “shocked, absolutely shocked” after seeing news of the deaths.
“It’s a quiet estate, things don’t happen around here like this,” he said, sharing that he had heard of the news on social media.
Meanwhile, neighbour Jane Scrafton told ITV News Anglia: “I saw all the police coming around this morning, and the ambulance and I thought that there was something going on. I thought it was quite serious for the amount of police.”
Gary Blundell, a Liberal Democrat councillor who represents New Costessey at South Norfolk District Council, said that residents and people would be offered “the support they need”.
He reassured locals that there was “no danger to the public” saying that the community had been left “stunned” by the incident.
He told the Eastern Daily Press: “It’s shocking – My heart goes out to anybody involved in this.
“Residents in Queen’s Hill are stunned – we will need to get anyone affected the support they need.”