  Let $G$ be a simple algebraic group over an algebraically closed field and
let $X$ be an irreducible subvariety of $G^r$ with $r \geqslant 2$. In this
paper, we consider the general problem of determining if there exists a tuple
$(x_1, \ldots, x_r) \in X$ such that $\langle x_1, \ldots, x_r \rangle$ is
Zariski dense in $G$. We are primarily interested in the case where $X = C_1
\times \cdots \times C_r$ and each $C_i$ is a conjugacy class of $G$ comprising
elements of prime order modulo the center of $G$. In this setting, our main
theorem gives a complete solution to the problem when $G$ is a symplectic or
orthogonal group. By combining our results with earlier work on linear and
exceptional groups, this gives a complete solution for all simple algebraic
groups. We also present several applications. For example, we use our main
theorem to show that many faithful representations of symplectic and orthogonal
groups are generically free. We also establish new asymptotic results on the
probabilistic generation of finite simple groups by pairs of prime order
elements, completing a line of research initiated by Liebeck and Shalev over 25
years ago.
