We present a new analysis of finite element methods for partial differential equations over curved domains. In many applications, a change of variables translates a physical Poisson problem over a curved physical domain into a parametric Poisson problem over a polytopal parametric domain. Whilst this change of variables greatly simplifies the geometry and numerical implementations, the coordinate transformation typically features only low regularity. In the parametric Poisson problem, this manifests as rough coefficients and data, which diminish the elliptic regularity, and as roughness of the parametric solution. Our main result addresses how to nevertheless recover high-order finite element convergence rates, the key component being a recently developed broken Bramble-Hilbert lemma. This analysis has numerous applications, where the geometric transformation is either computable or merely a theoretical tool. We propose a simplified technique as easier, more broadly applicable, yet just as powerful as previous isoparametric methods. In particular, we reassess the error analysis of isoparametric finite element methods and prove high-order error estimates for isoparametric FEM even when the physical solution is not continuous. Numerical examples confirm our theoretical predictions.& COPY; 2023 The Authors. Published by Elsevier B.V. on behalf of IMACS. This is an open access article under the CC BY license (http://creativecommons.org/licenses /by /4 .0/).