In recent years, Trotter formulas have emerged as a leading approach for simulating quantum dynamics on quantum computers, owing to their ability to exploit locality and commutator structure of the Hamiltonian. However, a major problem facing Trotter formulas is their inability to achieve poly-logarithmic scaling with the error tolerance. We address this problem by providing a well-conditioned extrapolation scheme that takes data from Trotter-Suzuki simulations obtained for specifically chosen Trotter step sizes and estimates the value that would be seen in the limit where the Trotter step size goes to zero. We show this leads, even for the first order Trotter formula, to Õ (1/ϵ) scaling for phase estimation and Õ (t2/ϵ) scaling for estimating time-evolved expectation values for simulation time t and error tolerance ϵ. This is better scaling with the error tolerance than the best known un-extrapolated Trotter formulas. Additionally, we provide a new approach for phase estimation that is unbiased and also provide a new approach for estimating the Trotter error on a quantum computer through extrapolation which yields a new way to independently assess the errors in a Trotter simulation.