(1) It is a theorem of Stallings [1978, "Constructions"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.
(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to "rotate" the former by 90 degrees. Then the given diagram can be converted without too high a "cost".
(3) There is an algorithm to convert a standard diagram into a braid closure. See the "Knot Book" by Adams.