Two Seifert surfaces of links in $S^3$ are said to be twist equivalent if one can
be obtained from the other, up to isotopy, by repeatedly performing operations consisting
of cutting along an embedded arc, applying a full twist near one copy of the arc, and
re-gluing. By using bridge spheres for their boundary links, we provide a new method of
distinguishing twist equivalence classes of Seifert surfaces of any given genus. Given a
Seifert surface $F$ of a link $L$, we show that the bridge numbers of the boundary links of
Seifert surfaces twist equivalent to $F$ are uniformly bounded above by a constant
depending only on $F$. By computing this constant in one simple case and applying a result
of Pfeuti, we deduce that $b(L)\leq 2g_c(L)+|L|$ for any link $L$ in $S^3$, where $g_c(L)$
denotes the canonical genus of $L$. Various consequences of this inequality are discussed.
We then apply the main theorem to show that there are infinitely many distinct twist
equivalence classes represented by free, genus one Seifert surfaces, answering a question
of Pfeuti in the negative.