Suppose the group $G =\langle X\mid \Delta\rangle$ (a presentation of $G$), where $\Delta$ is a set of reduced words in $X$. We have $G = F/R$ ($F$ is the free group generated by $X$),where $R$ is the normal subgroup of $F$ generated by $\Delta$.
Then the note I read says that $R$ is actually the set of all products of conjugates of elements of $\Delta \cup\Delta^{−1}$. Why? This doesn't make sense to me.
Say $X=\{a, b, c\}, \Delta=\{a^{2}\}$, then the products of conjugates of elements of $\Delta\cup \Delta^{−1}$ are all about $a$. By definition of normal subgroup, we should have $ba^{2}b^{-1} \in R$, which is obviously not the case, since $b$ can't appear in the products of conjugates of elements of $\Delta \cup \Delta ^{−1}$.
I think there is just some terminological confusion here. The BLAH generated by $X$ means the smallest BLAH that contains $X$ (and is a meaningful notion if the notion BLAH enjoys some sensible properties). So (with BLAH = "normal subgroup"), the normal subgroup generated by $X$ is the smallest normal subgroup containing $X$. This is bigger in general than the smallest subgroup containing $X$. The issue flagged in the title of your question does not apply to the document you cite, which is talking about the normal subgroup generated by $\Delta$ not the subgroup generated by $\Delta$.