  Let $F$ be a number field, $\pi$ either a unitary cuspidal automorphic
representation of $\mathrm{GL}(2)/F$ or a unitary Eisenstein series, and $\chi$
a unitary Hecke character of analytic conductor $C(\chi).$ We develop a
regularized relative trace formula to prove a refined hybrid subconvex bound
for $L(1/2,\pi\times\chi).$ In particular, we obtain the Burgess subconvex
bound \begin{align*}
L(1/2,\pi\times\chi)\ll_{\pi,F,\varepsilon}C(\chi)^{\frac{1}{2}-\frac{1}{8}+\varepsilon},
\end{align*} where the implied constant depends on $\pi,$ $F$ and
$\varepsilon.$
