  Monitored quantum circuits can exhibit an entanglement transition as a
function of the rate of measurements, stemming from the competition between
scrambling unitary dynamics and disentangling projective measurements. We study
how entanglement dynamics in non-unitary quantum circuits can be enriched in
the presence of charge conservation, using a combination of exact numerics and
a mapping onto a statistical mechanics model of constrained hard-core random
walkers. We uncover a charge-sharpening transition that separates different
scrambling phases with volume-law scaling of entanglement, distinguished by
whether measurements can efficiently reveal the total charge of the system. We
find that while R\'enyi entropies grow sub-ballistically as $\sqrt{t}$ in the
absence of measurement, for even an infinitesimal rate of measurements, all
average R\'enyi entropies grow ballistically with time $\sim t$. We study
numerically the critical behavior of the charge-sharpening and entanglement
transitions in U(1) circuits, and show that they exhibit emergent Lorentz
invariance and can also be diagnosed using scalable local ancilla probes. Our
statistical mechanical mapping technique readily generalizes to arbitrary
Abelian groups, and offers a general framework for studying
dissipatively-stabilized symmetry-breaking and topological orders.
