  Dirty fireballs are a hypothesized class of relativistic massive-star
explosions with an initial Lorentz factor $\Gamma_\mathrm{init}$ below the
$\Gamma_\mathrm{init}\sim100$ required to produce a long-duration gamma-ray
burst (LGRB), but which could still produce optical emission resembling LGRB
afterglows. Here we present the results of a search for on-axis optical
afterglows using the Zwicky Transient Facility (ZTF). Our search yielded seven
optical transients that resemble on-axis LGRB afterglows in terms of their red
colors ($g-r>0$ mag), faint host galaxy ($r>23$ mag), and rapid fading
($dr/dt>1$ mag/day). Spectroscopy of the transient emission within a few days
of discovery established cosmological distances ($z=0.876$ to $z=2.9$) for six
events, tripling the number of afterglows with redshift measurements discovered
by optical surveys without a $\gamma$-ray trigger. Upon a retrospective search,
four events (ZTF20abbiixp/AT2020kym, ZTF21aagwbjr/AT2021buv,
ZTF21aakruew/AT2021cwd, ZTF21abfmpwn/AT2021qbd) turned out to have a likely
associated LGRB (GRB200524A, GRB210204A, GRB210212B, GRB210610B), while three
did not (ZTF20aajnksq/AT2020blt, ZTF21aaeyldq/AT2021any,
ZTF21aayokph/AT2021lfa). Our search revealed no definitive new class of events:
the simplest explanation for the apparently "orphan" events is that they were
regular LGRBs missed by high-energy satellites due to detector sensitivity and
duty cycle, although it is possible that they were intrinsically faint in
$\gamma$-rays or viewed slightly off-axis. We rule out a scenario in which
dirty fireballs have a similar energy per solid angle to LGRBs and are an order
of magnitude more common. In addition, we set the first direct constraint on
the ratio of the opening angles of the material producing $\gamma$-rays and the
material producing early optical afterglow emission, finding that they must be
comparable.
