$\DeclareMathOperator\SL{SL}$The Cartan–Dieudonné theorem states that each element $g \in \operatorname{O}(V)$, where $V$ is a quadratic space of dimension $n$ over a field of characteristic $\neq 2$, can be written as a product of $\leq n$ reflections.
Something similar is true for $\SL_n(k)$ for $\operatorname{char}k \neq 2$: each element $g$ can be written as a product of elements of order $4.$ Indeed, it suffices to prove this for $n=2$. Then $s_t:=\begin{pmatrix} 0 & t \\ -\frac 1t & 0 \end{pmatrix}$ is of order $4$ . Let $h(t):=s_ts_{-1}=\begin{pmatrix} t & 0 \\ 0 & \frac 1t \end{pmatrix}$, then each element in $U$, the group of upper triangular matrices, is a product of two conjugates of elements $h(t),h(t')$ (provided $|k|\geq 4$). Similarly for the lower triangular matrices $V$, and then $\SL_2$ is generated by these subgroups.
Since a simply connected semisimple $k$-split group is generated by $\SL_2$'s, the same argument applies.
What can we say about other algebraic groups? Clearly unipotent groups (in characteristic 0) don't have elements of finite order, so nothing there.
What about (the $k$-rational points of) anisotropic groups?
$s_t$
is not a "reflection" and no bounds on number of generators are implicit here. The type of field (finite, etc.) plays a role, along with the$k$
-structure. Generation of semisimple groups comes up in work of Steinberg, Tits, and others; for more general groups you want to distinguish cases where a Levi decomposition exists and those in characteristic$p$
where it doesn't, etc. $\endgroup$