The aim of this work is to prove global $L^p$ Carleman estimates for the Laplace operator in dimension $d \geq 3$. Our strategy relies on precise Carleman estimates in strips, and a suitable gluing of local and boundary estimates obtained through a change of variables. The delicate point and most of the work thus consists in proving Carleman estimates in the strip with a linear weight function for a second order operator with coefficients depending linearly on the normal variable. This is done by constructing an explicit parametrix for the conjugated operator, which is estimated through the use of Stein Tomas restriction theorems. As an application, we deduce quantified versions of the unique continuation property for solutions of $\Delta u = V u + W_1 \cdot \nabla u + \div(W_2 u)$ in terms of the norms of $V$ in $L^{q_0}(\Omega)$, of $W_1$ in $L^{q_1}(\Omega)$ and of $W_2$ in $L^{q_2}(\Omega)$ for $q_0 \in (d/2, \infty]$ and $q_1$ and $q_2$ satisfying either $q_1, \, q_2 > (3d-2)/2$ and $1/q_1 + 1/q_2< 4 (1-1/d)/(3d-2)$, or $q_1, \, q_2 > 3d/2$.