Police said two rival groups were involved in a series of clashes in the Darnall area of the city before a red hatchback vehicle was deliberately steered off the road at speed onto the footpath, mowing down the teenager.
He was rushed to hospital after the incident in Coleridge Road at 7.35pm on Thursday but died from multiple injuries.
Police were expecting to officially identify the victim, believed to be an 18-year-old man, today.
Chief Superintendent Simon Torr, police commander for Sheffield, said: “Four men have been arrested on suspicion of murder and are currently in custody.
“There were, we believe, a couple of other incidents prior to this, involving disorder and young people in the Darnall area.
“We are looking at these incidents to see if they are linked and, if so, how they are linked.”
It is thought that the killing was the latest in a long-running feud between rival groups of teenagers in the area, where police have posted notices in an effort to prevent large numbers of youths congregating at night.
The suburb has been branded a “no go” area by residents who met South Yorkshire Police representatives last month after a string of serious incidents.
They included the rape of a 41-year-old woman by two men and the stabbing of a 28-year-old man. Residents also claim there is prostitution, drug dealing and gangs of youths roaming the streets, intimidating local residents and forcing some to “live as prisoners in their own homes.”
One man, who was at the scene shortly after Thursday’s killing, said: “You couldn’t tell whether he had been run over because he was laid at the side of some railings.
“Whether somebody had moved him there or not I don’t know. I didn’t see him get knocked over.
“I just saw him at the side of the road but I could see it was serious because there was so much blood.”
Another local resident, who did not wish to be identified, said: “There has been one gang which has had a problem with another gang in the area for some time.
“On the night this happened there were more fights and then someone jumped into a car and drove onto the pavement and rammed this lad.”
Police sealed off Coleridge Gardens and Coleridge Road following the incident, as forensic teams combed the area and examined the hatchback car at the side of the road.
Cheryl Barber, who lives in Darnall, added: “Every few weeks something happens around here and it just seems to be never ending. There have been rapes and murders and drug dealing.
“Sometimes you just want to stop in your house instead of going out because of what is happening on the streets.”
Mr Torr said that the killing was not thought to be connected with Sheffield’s notorious “postcode” gangs and added: “If we’re talking about organised criminal gangs, S3 and S4, then no it’s not gang related.
“We’re unsure as to what the precursory incidents are. There was tension and disorder between groups of youths, but these are not formal gangs – they are groups of youths who have a grievance with each other.”
He added: “We’ve had 50 community leaders in at the police station, to give them the information we can give them and stop speculation.
“Clearly, with the death of any young person, people are upset and there’s tension. We’ve appealed to people to be calm and to let us investigate, not take matters into their own hands.
“I’d also like to offer my condolences to the victim’s family at this tragic time.”