A flexible screen support device and a flexible display terminal are provided. The flexible screen support device includes: a rotating assembly including a rotation shaft; a first housing and a second housing respectively connected to the rotating assembly; and a fixing piece, located on one side, away from the rotating assembly, of the first housing and the second housing. At least one of the first housing and the second housing rotates around the rotating assembly to make the fixing piece bend or flatten, and a horizontal length of a side surface, perpendicular to the rotation shaft, of the fixing piece in a flattened state is equal to a length of the side surface, which is a curved surface, of the fixing piece in a bent state.