Other refugees who survived the accident told reporters on Friday that Abdullah Kurdi was piloting the boat which capsized, contradicting Kurdi's claim that he was traveling on the boat as a passenger, and that the captain panicked and jumped overboard when the vessel hit a wave.
The version of events given by Abbas, her husband, and their one daughter who survived the trip, has been backed up by another Iraqi survivor.
At least 12 people are reported to have died when the boat, a 4.5 meter rubber dinghy, capsized in the Aegean Sea trying to reach the Greek coast.
Kurdi was driving the boat and "going crazy, like speed," Abbas told media in Australia, adding that after the boat capsized, when its passengers were in the water, Kurdi asked them not to identify him as the driver, saying "please don't give me away."
"This is not true. If I was a people smuggler, why would I put my family in the same boat as the other people? I paid the same amount to the people smugglers," said Kurdi.
The image of Aylan Kurdi's dead body being carried ashore in Greece brought worldwide attention to the plight of refugees. His father Abdullah, who also lost his wife and five-year-old son Galip in the tragedy, has since returned to the family's home in Kobane, Syria.