  We study smooth isotopy classes of complex curves in complex surfaces from
the perspective of the theory of bridge trisections, with a special focus on
curves in $\mathbb{CP}^2$ and $\mathbb{CP}^1\times\mathbb{CP}^1$. We are
especially interested in bridge trisections and trisections that are as simple
as possible, which we call "efficient". We show that any curve in
$\mathbb{CP}^2$ or $\mathbb{CP}^1\times\mathbb{CP}^1$ admits an efficient
bridge trisection. Because bridge trisections and trisections are nicely
related via branched covering operations, we are able to give many examples of
complex surfaces that admit efficient trisections. Among these are
hypersurfaces in $\mathbb{CP}^3$, the elliptic surfaces $E(n)$, the Horikawa
surfaces $H(n)$, and complete intersections of hypersurfaces in
$\mathbb{CP}^N$. As a corollary, we observe that, in many cases, manifolds that
are homeomorphic but not diffeomorphic have the same trisection genus, which is
consistent with the conjecture that trisection genus is additive under
connected sum. We give many trisection diagrams to illustrate our examples.
