  In this article, we study the enumeration by length of several walk models on
the square lattice. We obtain bijections between walks in the upper half-plane
returning to the $x$-axis and walks in the quarter plane. A recent work by
Bostan, Chyzak, and Mahboubi has given a bijection for models using small
north, west, and south-east steps. We adapt and generalize it to a bijection
between half-plane walks using those three steps in two colours and a
quarter-plane model over the symmetrized step set consisting of north,
north-west, west, south, south-east, and east. We then generalize our
bijections to certain models with large steps: for given $p\geq1$, a bijection
is given between the half-plane and quarter-plane models obtained by keeping
the small south-east step and replacing the two steps north and west of length
1 by the $p+1$ steps of length $p$ in directions between north and west. This
model is close to, but distinct from, the model of generalized tandem walks
studied by Bousquet-M\'elou, Fusy, and Raschel.
