We study stochastic linear bandits with heavy-tailed rewards, where the rewards have a finite $(1+ε)$-absolute central moment bounded by $\upsilon$ for some $ε\in (0,1]$. We improve both upper and lower bounds on the minimax regret compared to prior work. When $\upsilon = \mathcal{O}(1)$, the best prior known regret upper bound is $\tilde{\mathcal{O}}(d T^{\frac{1}{1+ε}})$. While a lower with the same scaling has been given, it relies on a construction using $\upsilon = \mathcal{O}(d)$, and adapting the construction to the bounded-moment regime with $\upsilon = \mathcal{O}(1)$ yields only a $Ω(d^{\fracε{1+ε}} T^{\frac{1}{1+ε}})$ lower bound. This matches the known rate for multi-armed bandits and is generally loose for linear bandits, in particular being $\sqrt{d}$ below the optimal rate in the finite-variance case ($ε= 1$). We propose a new elimination-based algorithm guided by experimental design, which achieves regret $\tilde{\mathcal{O}}(d^{\frac{1+3ε}{2(1+ε)}} T^{\frac{1}{1+ε}})$, thus improving the dependence on $d$ for all $ε\in (0,1)$ and recovering a known optimal result for $ε= 1$. We also establish a lower bound of $Ω(d^{\frac{2ε}{1+ε}} T^{\frac{1}{1+ε}})$, which strictly improves upon the multi-armed bandit rate and highlights the hardness of heavy-tailed linear bandit problems. For finite action sets, we derive similarly improved upper and lower bounds for regret. Finally, we provide action set dependent regret upper bounds showing that for some geometries, such as $l_p$-norm balls for $p \le 1 + ε$, we can further reduce the dependence on $d$, and we can handle infinite-dimensional settings via the kernel trick, in particular establishing new regret bounds for the Matérn kernel that are the first to be sublinear for all $ε\in (0, 1]$.