The outbound Saab 340 aircraft, operated by Scottish regional airline Loganair, was due to fly to Glasgow from Stornoway Airport on Isle of Lewis at 8.33am local time when the accident happened.
The Scottish Ambulance Service said two people were taken to Western Isles Hospital with minor injuries. Police said that these two were male.
Two female passengers also sustained minor injuries but declined medical assistance.
In a statement, Stornoway Airport said: "Efforts to remove the aircraft from the runway are ongoing and investigations into the cause of the incident are under way."
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Stornoway Airport said: "It is our intention to resume normal operations as soon as the aircraft is safely removed from the runway."
The Scottish Fire and Rescue Service sent three fire engines to the scene.
After leaving the runway the plane came to rest on the grass next to it.
A Police Scotland spokesman said: "A full air accident investigation has now been launched and will be carried out by the Air Accidents Investigation Branch in conjunction with Police Scotland."
A passenger told the BBC it felt as if the tail was caught by a crosswind.