Catalyzes the reaction that attaches fucose through an O-glycosidic linkage to a conserved serine or threonine residue in the consensus sequence C1-X(2,3)-S/T-C2-X2-G of thrombospondin type 1 repeats where C1 and C2 are the first and second cysteines, respectively. O-fucosylates members of several protein families including the ADAMTS family, the thrombosporin (TSP) and spondin families. The O-fucosylation of TSRs is also required for restricting epithelial to mesenchymal transition (EMT), maintaining the correct patterning of mesoderm and localization of the definite endoderm (By similarity). Required for the proper secretion of ADAMTS family members such as ADAMSL1 and ADAMST13.