We report a measurement of the E-mode polarization power spectrum of the
cosmic microwave background (CMB) using 150 GHz data taken from July 2014 to
December 2016 with the POLARBEAR experiment. We reach an effective polarization
map noise level of $32\,\mu\mathrm{K}$-$\mathrm{arcmin}$ across an observation
area of 670 square degrees. We measure the EE power spectrum over the angular
multipole range $500 \leq \ell <3000$, tracing the third to seventh acoustic
peaks with high sensitivity. The statistical uncertainty on E-mode bandpowers
is $\sim 2.3 \mu {\rm K}^2$ at $\ell \sim 1000$ with a systematic uncertainty
of 0.5$\mu {\rm K}^2$. The data are consistent with the standard $\Lambda$CDM
cosmological model with a probability-to-exceed of 0.38. We combine recent CMB
E-mode measurements and make inferences about cosmological parameters in
$\Lambda$CDM as well as in extensions to $\Lambda$CDM. Adding the ground-based
CMB polarization measurements to the Planck dataset reduces the uncertainty on
the Hubble constant by a factor of 1.2 to $H_0 = 67.20 \pm 0.57 {\rm km\,s^{-1}
\,Mpc^{-1}}$. When allowing the number of relativistic species ($N_{eff}$) to
vary, we find $N_{eff} = 2.94 \pm 0.16$, which is in good agreement with the
standard value of 3.046. Instead allowing the primordial helium abundance
($Y_{He}$) to vary, the data favor $Y_{He} = 0.248 \pm 0.012$. This is very
close to the expectation of 0.2467 from Big Bang Nucleosynthesis. When varying
both $Y_{He}$ and $N_{eff}$, we find $N_{eff} = 2.70 \pm 0.26$ and $Y_{He} =
0.262 \pm 0.015$.