A bath curtain rack structure includes a curved transverse rod, two pivot members and two wall-mounted brackets. The curved transverse rod has two ends which are connected to hollow pipe portions of the pivot members, respectively. Each pivot member includes a flat rounded end with a semi-circle flange at one end thereof. Each wall-mounted bracket includes a central protruding post which is inserted in a dual awl-shaped guide hole of the pivot member. The wall-mounted bracket further has a pair of lock holes for insertion of screws to be secured on a wall, a recess corresponding in shape to the flat round end, two stoppers to stop the semi-circle flange, and a protruding opening to receive the flat rounded end and the semi-circle flange of the pivot member. A second headless screw is screwed in a second threaded hole of the flat rounded end to engage with the concave pivot point. A first headless screw is screwed in a first threaded hole of the pivot member to engage with the ends of the curved transverse rod.