= Solution
Fix a <Borel subgroup> $B$ and a <maximal algebraic torus> $T\subset B$. Conjugacy of <Borel subgroups> and the equality $N_G(B)=B$ identify the <flag variety> with $G/B$, by $gB\mapsto{}^gB$.
Every simultaneous conjugacy orbit on pairs contains a pair $(B,{}^gB)$: first conjugate its first member to $B$. The remaining allowed conjugations are by $B$. Two such pairs are in the same orbit exactly when $g'$ belongs to $BgB$. Indeed ${}^{g'}B={} ^{bg}B$ is equivalent to $(bg)^{-1}g'\in N_G(B)=B$. Thus there is an explicit bijection
$$
\boxed{G\backslash(\mathcal B\times\mathcal B)\ \longleftrightarrow\ B\backslash G/B.}
$$
The <Bruhat decomposition> is the disjoint partition $G=\coprod_{w\in W}B\dot wB$, with $W=N_G(T)/T$. Under the displayed bijection its double coset $B\dot wB$ becomes the orbit
$$
O(w)=G\cdot(B,{}^{\dot w}B).
$$
This proves that the orbits are indexed by the <Weyl group>. Conversely, the assertion that these particular $O(w)$ exhaust the orbits without repetitions says precisely that the $B\dot wB$ exhaust $G$ without repetitions. Hence the orbit description and the <Bruhat decomposition> are equivalent, not merely two sets of the same cardinality. Their common index is the <relative position of Borel subgroups>.
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