Any generic planar closed curve with n crossings can be turned into a simple closed curve by applying O(n3/2) homotopy moves without ever increasing the number of self-crossings; this improves over the O(n2) upper bound from Steinitz [Ency. Math. Wiss. III 1916], and matches the best lower bound. We prove the existence of a positive move that decreases the depth-sum potential at every step. Using similar techniques, we show that any 2-terminal plane graph with n vertices can be reduced to a single edge between the terminals using O(n3/2) electrical transformations, consisting of degree-1 reductions, series-parallel reductions, and ΔY-transformations; this proves a conjecture of Feo and Provan that was open for more than 30 years.