  We investigate the structure of a variety of new Moishezon twistor spaces, by
utilizing the pluri-half-anti-canonical map from the twistor spaces. Each of
these twistor spaces is bimeromorphic to a double covering of a scroll of
planes over a rational normal curve, and the branch divisor of the double cover
is a cut of the scroll by a quartic hypersurface. In particular, the double
covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the
twistor spaces have a pencil of rational surfaces with big anti-canonical
class. The base locus of the last pencil is a cycle of rational curves, and it
is an anti-canonical curve on smooth members of the pencil.
  These twistor spaces are naturally classified into four types according to
the type of singularities of the branch divisor, or equivalently, those of the
Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface
satisfies a strong constraint and as a result the defining polynomial of the
quartic hypersurface has to be of a specific form.
  Together with our previous result, the present result completes a
classification of Moishezon twistor spaces whose half-anti-canonical system is
a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil
have been understood for a long time before. In the opposite direction, no
example is known of a Moishezon twistor space whose half-anti-canonical system
is smaller than a pencil.
