Berdymukhamedov suffered the fall seconds after he won a kilometre-long horse race in Ashgabat on his favourite mount Berkarar (The Powerful) at an annual horse festival on Sunday celebrating the elite Turkmen Akhal-Teke breed.
The videos show a man identified as the president and dressed in traditional riding robes being thrown to the ground as the horse appears to stutter just after crossing the finishing line.
Eventually a van appears and takes him away. The video goes on to show a figure resembling Berdymukhamedov later waving to the applauding crowd in an apparent bid to show all is well.
Turkmenistan state media this week reported in great detail on the president's victory in the horse race, which won him a cash prize of $11 million, all of which he donated to an official horse fund.
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Government newspaper Neutral Turkmenistan devoted four pages to his victory in a report headlined "The Striving of the Nation, The Striving of the Leader", with six giant pictures of the president in national costume.
"The race gave the impression that the horses were flying from the earth, as if their hooves were not even touching the ground!" gushed the report in Neutral Turkmenistan.
The paper quoted the president as saying that "sitting on the horse I acutely felt the pride of the people, which had the chance today to share its love for the Akhal-Teke."
The president was shown on state television Monday evening -- apparently in good health -- holding a video conference with regional governors. Yesterday he was shown holding a meeting on migration.
The Gundogar news site said officials at Ashgabat's international airport had been making intensive checks of passengers leaving the country in search of phones, tablets and cameras with footage of the incident.
It quoted sources at the interior ministry as saying several dozen people have already been arrested on suspicion of trying to smuggle out footage.
However the incident was shown on Russian state television. Sources inside Turkmenistan said today that access to Russian social networks had been halted and all the Internet was working painfully slowly.