Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle $$G\to E\to B,$$ then $B=E/G$, the orbit space under action of $G$. Let $BG$ be the classifying space of $G$.
How to get the following fiber sequence: $$G\simeq \Omega BG\to E\to B\to BG?$$
The classifying space $BG$ has a universal principal $G$-bundle $\xi_{u} : EG \to BG$ with the universal property that for any principal $G$-bundle $\xi : E \to B$, there exists a classifying map (unique up to homotopy) $f : B \to BG$ such that $f^* \xi_u = \xi$. In other words there is a pullback diagram: $$\require{AMScd} \begin{CD} E @>>> EG \\ @V{\xi}VV @V{\xi_u}VV \\ B @>{f}>> BG \end{CD}$$ It follows that $EG$ is contractible, so $E$ is the homotopy pullback $B \times^h_{BG} *$. Thus you get a fiber sequence (I don't know how you defined them but that's probably equivalent to this) $E \to B \to BG$.