This is a very classical topic in algebraic group theory. If $k$ is a field and $G$ is an quasi-simple algebraic group defined over $k$ which is $k$-isotropic, then the group $G^+\subseteq G(k)$ generated by the unipotent elements in the $k$-parabolic subgroups is 'in most cases' simple modulo its center. (An algebraic group is quasi-simple if it has no proper connected normal subgroups defined over $k$, and $k$-isotropic if it has a $k$-split torus.)
The precise result, due to J. Tits, is as follows [J. Tits, Algebraic and abstract simple groups, Ann. Math. 80 (2), 1964]. If $G$ is a quasi-simple $k$-isotropic algebraic group defined over a field $k$ with at least 4 elements, then $G^+/Z(G^+)$ is simple. (Tits explains in the article also the exceptional cases over fields with 2 or 3 elements. Exceptions occur only in $k$-rank 1 or 2).
This reduces the question to the quotient $G(k)/G^+$. The computation of this quotient is known as the Kneser-Tits problem. If $G$ is simply connected, then the quotient is in many cases trivial. This quotient is also called the Whitehead group $W(G,k)=G(k)/G^+$. If $G$ is simply connected and split or quasi-split, then $W(G,k)=1$ [Tits, Groupes de Whitehead de groupes algebriques sur un corps, Sem. Bourbaki vol. 1976/77]. The Whitehead group is also known to be trivial for certain fields (such as $\mathbb R$), or for certain types of groups.
In the example $G=SL_n$ over any field $k$ with enough elements we have $G(k)=G^+$ (the typical unipotents being the conjugates of the upper triangular matrices with 1 on the diagonal) and indeed $SL_n(k)/Z(SL_n(k))$ is simple (and $SL_n$ is simply connected as an algebraic group). Thus $W(SL_n,k)=1$.
If $G$ is not simply connected, the answer is more complicated. The adjoint group of $SL_n$ is the quasi-simple group $G=PGL_n$, which is not simply connected. In this case $PGL_n(k)^+=PSL_n(k)=SL_n(k)/Z(SL_n(k))$. The quotient $PGL_n(k)/PSL_n(k)=G(k)/G^+$ is isomorphic to $k^*/(k^*)^n$.
[Note: the map $SL_n(k)\to PGL_n(k)$ is not surjective on the $k$-rational points. The terminology I use here differs from YCor's notation above, but I think it is in accordance with many books on algebraic groups, such as Borel or Milne.]
So for simply connected $k$-isotropic quasi-simple algebraic groups, Tits' Theorem reduces the question to the computation of the Whitehead group.
If the group $G$ is not $k$-isotropic, then $G(k)$ may have many nontrivial normal subgroups. Consider for example the special orthogonal group $G=SO_n$ for the standard bilinear form. For $n\geq 5$, this group is quasi-simple. For $k=\mathbb R$, the compact Lie group $G(k)$ is simple modulo its center. But if $k$ is a non-archimedean real closed field (eg. the field of nonstandard reals $^*\mathbb R$), then the matrices infinitesimally close to 1 generate a normal subgroup in $G(k)$.