  2D materials based superlattices have emerged as a promising platform to
modulate band structure and its symmetries. In particular, moir\'e periodicity
in twisted graphene systems produces flat Chern bands. The recent observation
of anomalous Hall effect (AHE) and orbital magnetism in twisted bilayer
graphene has been associated with spontaneous symmetry breaking of such Chern
bands. However, the valley Hall state as a precursor of AHE state, when
time-reversal symmetry is still protected, has not been observed. Our work
probes this precursor state using the valley Hall effect. We show that broken
inversion symmetry in twisted double bilayer graphene (TDBG) facilitates the
generation of bulk valley current by reporting the first experimental evidence
of nonlocal transport in a nearly flat band system. Despite the spread of Berry
curvature hotspots and reduced quasiparticle velocities of the carriers in
these flat bands, we observe large nonlocal voltage several micrometers away
from the charge current path -- this persists when the Fermi energy lies inside
a gap with large Berry curvature. The high sensitivity of the nonlocal voltage
to gate tunable carrier density and gap modulating perpendicular electric field
makes TDBG an attractive platform for valley-twistronics based on flat bands.
