The traffic assignment problem is a widely used formulation for designing, analyzing, and evaluating transportation networks. The inputs to this model, besides the network topology, are the origin–destination (OD) demand matrix and travel latency cost functions. It has been observed that small perturbations to these inputs have a large impact on the solution. However, most efforts on estimating these using data do so separately and are typically based on parametric models or surveys. In this article, we present a kernel-based framework that jointly estimates the OD demand matrix and the travel latency function in single- and multiclass vehicle networks. To that end, we formulate a bilevel optimization problem, and then, we transform it to a quadratically constrained quadratic program (QCQP). To solve this QCQP, we propose a trust-region feasible direction algorithm that sequentially solves a quadratic program. In addition, we also provide an alternating optimization method. Our results show that the QCQP method achieves better estimates when compared with disjoint and sequential methods. We show the applicability of the method by performing case studies using data for the transportation networks of Eastern Massachusetts Area and New York City.