Coward and Lackenby have an upper bound on the number of Reidemeister moves, which is a tower of exponentials. The existence of some such bound is not surprising, since Waldhausen had proven that the knot isotopy problem was solvable, so some computable upper bound exists.

Suppose you had a much better upper bound on the number of crossings of diagrams in the sequence of moves than their bound. Then since the number of diagrams with $c$ crossings is no more than say $k^{k^c}$ for some $k$, one would get a much better bound on the number of reidemeister moves to get between two diagrams. So I think one would need a new idea to get such an estimate.

Coward and Lackenby have an upper bound on the number of Reidemeister moves, which is a tower of exponentials. The existence of some such bound is not surprising, since Waldhausen had proven that the knot isotopy problem was solvable, so some computable upper bound exists.

Suppose you had a much better upper bound on the number of crossings of diagrams in the sequence of moves than their bound. Then since the number of diagrams with $c$ crossings is no more than say $k^{k^c}$ for some $k$, one would get a much better bound on the number of reidemeister moves to get between two diagrams. So I think one would need a new idea to get such an estimate.