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Adam Lewis said the impact of the crash was "overwhelming" but said he was just thankful his family were still alive.
"You don't expect a big white van to come and smack you off the pavement," he told Sky News, adding, "Why someone would ever want to do that I will never know."
Mr Lewis said "our hearts go out" to the family of Karina Menzies, who was killed in a separate attack during the hit-and-run rampage, and admitted: "We are so relieved we're still ok."
Mr Lewis said his "brave" partner will have an operation to put pins in her leg after breaking her fibula and tibula in the crash, while a hairline fracture in her ankle will be strapped. She also wounded her left hip. He added that his two-year-old "had been incredibly brave."
Adam Lewis, who saw his partner and baby daughter hit by the rampaging van in Cardiff, has spoken of the moment the vehicle smashed into his family.
With two-year-old Amelia May sat next to him bearing heavy facial scars from the collision, Mr Lewis told Sky News his parter Annie was pushing their daughter's pushchair on the pavement when "all of a sudden we got hit behind by the van."
He added: "It was too late he just smashed right into us."
The father, who was knocked to the floor by the van's wing mirror, said: "Unfortunately my partner and our baby bore the brunt of it."
He said Ms Lewis "buckled" when the vehicle struck and "just flew" with her and Amelia "catapulted" into the air.
The family of Karina Menzies, who was killed in a hit-and-run van attack in Cardiff, have released the following statement:
"Karina was a lovely girl. She lived for her kids and her brothers and sisters and everything she did was for her family. She was a good friend to everyone and was well thought of in the community. She will be sorely missed by everyone."
The brother of the woman killed in a hit-and-run attack in Cardiff said his sister died trying to save the lives of her children.
Craig Menzies described the moment the van careered towards Karina Menzies and her children. He told Radio 5 Live:
Mr Menzies said the attack was "a waste of a life" and said his sister was "the world to me".
Police have been granted a further 36 hours to detain and question the suspect.
Detectives are calling for witnesses following a series of hit-and-run incidents in Cardiff on Friday.
A 31-year-old man has been arrested on suspicion of murder.
As police continue to question a man on suspicion of murder after yesterdays hit and runs attacks the community has remained shocked today over the atrocity.
ITV News Correspondent Emily Morgan reports:
Rev Jan Gould is the vicar at the Church of the Resurrection in Ely said that the Church had been open for locals to help deal with emotions after the hit and run attacks across Cardiff:
Tomorrow, a special service will be held at the church at 5pm, to provide "an opportunity for the whole community to come together, share their grief and find support".
Reverend Jan Gould, who will lead a service tomorrow for the victims and families of the town's fatal hit-and-run collisions in Ely's Church of the Resurrection, said the feeling in the town was "like a state of carnage".
Latest ITV News reports
Hundreds gathered to share their grief at a sombre church and prayer service after a hit-and-run rampage left one mother dead.
The shocked community of Ely in Cardiff will gather to pray for the victims of Friday's terrible hit-and-run attacks in the Welsh capital.