Catalyzes the reaction that attaches fucose through an O-glycosidic linkage to a conserved serine or threonine residue in the consensus sequence C1-X-X-S/T-C2 of thrombospondin type I repeats (TSRs) where C1 and C2 are the first and second cysteines of the repeat, respectively. O-fucosylates members of several protein families including the ADAMTS, the thrombospondin (TSP) and spondin families. Required for the proper secretion of ADAMTS family members such as ADAMTSL1 and ADAMTS13 (By similarity). 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). GDP-beta-L-fucose + L-seryl-[protein] = 3-O-(alpha-L-fucosyl)-L-seryl-[protein] + GDP + H(+) GDP-beta-L-fucose + L-threonyl-[protein] = 3-O-(alpha-L-fucosyl)-L-threonyl-[protein] + GDP + H(+) Protein modification; protein glycosylation. Mainly located in the endoplasmic reticulum. Belongs to the glycosyltransferase 68 family.