The invention relates to a thread cutting insert (1) comprising a upper face (20) and a lower face (21), at least three lateral walls (31, 32, 33, 34) extending between the upper face (20) and the lower face (21), and at least two cutting edges (41, 42, 43, 44) formed in a transition region between two of the lateral walls (31, 32, 33, 34) and the upper face (20), wherein at least one of the cutting edges (41, 42, 43, 44) is sandwiched between a first supporting surface (71a, 72a, 73a, 74a) and a second supporting surface (71 b, 72b, 73b, 74b) provided on the respective lateral wall (31, 32, 33, 34). Further, the invention relates to a tool holder for receiving a thread cutting insert (1).