  Geometrical frustration in strongly correlated systems can give rise to a
plethora of novel ordered states and intriguing magnetic phases, such as
quantum spin liquids. Promising candidate materials for such phases can be
described by the Hubbard model on an anisotropic triangular lattice, a
paradigmatic model capturing the interplay between strong correlations and
magnetic frustration. However, the fate of frustrated magnetism in the presence
of itinerant dopants remains unclear, as well as its connection to the doped
phases of the square Hubbard model. Here we investigate the local spin order of
a Hubbard model with controllable frustration and doping, using ultracold
fermions in anisotropic optical lattices continuously tunable from a square to
a triangular geometry. At half-filling and strong interactions $U/t \sim 9$, we
observe at the single-site level how frustration reduces the range of magnetic
correlations and drives a transition from a collinear N\'eel antiferromagnet to
a short-range correlated 120$^{\circ}$ spiral phase. Away from half-filling,
the triangular limit shows enhanced antiferromagnetic correlations on the
hole-doped side and a reversal to ferromagnetic correlations at particle
dopings above 20%, hinting at the role of kinetic magnetism in frustrated
systems. This work paves the way towards exploring possible chiral ordered or
superconducting phases in triangular lattices and realizing t-t' square lattice
Hubbard models that may be essential to describe superconductivity in cuprate
materials.
