  Microscopically understanding competing orders in strongly correlated systems
is a key challenge in modern quantum many-body physics. For example, the study
of magnetic polarons and their relation to pairing in the Fermi-Hubbard model
in different geometries remains one of the central questions, and may help to
understand the mechanism underlying unconventional superconductivity in
cuprates or transition metal dichalcogenides. With recent advances in analog
quantum simulation of the Fermi-Hubbard model on non-bipartite lattices,
frustrated physics can now be explored experimentally in systems lacking
particle-hole symmetry. Here, we study the singly doped $t-J$ model on the
triangular lattice, focusing on the competition between kinetic and magnetic
frustration as a function of temperature. In doublon doped systems, we uncover
a crossover between Nagaoka-type ferromagnetic (FM) correlations at high
temperature and exchange mediated antiferromagnetic (AFM) order around the
doublon at low temperature. For hole doped systems, kinetic
Haerter-Shastry-type AFM at high temperature as well as exchange interactions
at low temperature favor $120^{\circ}$ order, strengthening magnetic
correlations compared to the undoped system. In the ground state, the presence
of AFM correlations throughout a wide range of interactions indicates
confinement of both types of dopants. In this regime we firmly establish the
presence of linear confining potentials via energy scaling arguments,
supporting the picture of geometric strings in the frustrated triangular $t-J$
model.
