We advance the study of incentivized bandit exploration, in which arm choices are viewed as recommendations and are required to be Bayesian incentive compatible. Recent work of Sellke-Slivkins (Operations Research 2022) has shown that for the special case of independent arms, after collecting enough initial samples, the popular Thompson sampling algorithm becomes incentive compatible. This was generalized to the combinatorial semibandit in Hu-Ngo-Slivkins-Wu (NeurIPS 2022). We give an analog of this result for linear bandits, where the independence of the prior is replaced by a natural convexity condition. This opens up the possibility of efficient and regret-optimal incentivized exploration in high-dimensional action spaces. In the semibandit model, we also improve the sample complexity for the pre-Thompson sampling phase of initial data collection.