Pulsed optomechanical measurements enable squeezing, nonclassical state creation, and backaction-free sensing. We demonstrate pulsed measurement of a cryogenic nanomechanical resonator with record precision close to the quantum regime. We use these to prepare thermally squeezed and purified conditional mechanical states, and to perform full state tomography. These demonstrations exploit large vacuum optomechanical coupling in a nanophotonic cavity to reach a single-pulse imprecision of 9 times the mechanical zero-point amplitude x_{zpf}. We study the effect of other mechanical modes that limit the conditional state width to 58x_{zpf}, and show how decoherence causes the state to grow in time.