  In 2015, Kawarabayashi and Kreutzer proved the directed grid theorem
confirming a conjecture by Reed, Johnson, Robertson, Seymour, and Thomas from
the mid-nineties. The theorem states the existence of a function $f$ such that
every digraph of directed tree-width $f(k)$ contains a cylindrical grid of
order $k$ as a butterfly minor, but the given function grows non-elementarily
with the size of the grid minor.
  In this paper we present an alternative proof of the directed grid theorem
which is conceptually much simpler, more modular in its composition and also
improves the upper bound for the function $f$ to a power tower of height 22.
  Our proof is inspired by the breakthrough result of Chekuri and Chuzhoy, who
proved a polynomial bound for the excluded grid theorem for undirected graphs.
We translate a key concept of their proof to directed graphs by introducing
\emph{cycles of well-linked sets (CWS)}, and show that any digraph of high
directed tree-width contains a large CWS, which in turn contains a large
cylindrical grid, improving the result due to Kawarabayashi and Kreutzer from
an non-elementary to an elementary function.
  An immediate application of our result is an improvement of the bound for
Younger's conjecture proved by Reed, Robertson, Seymour and Thomas (1996) from
a non-elementary to an elementary function. The same improvement applies to
other types of Erd\H{o}s-P\'osa style problems on directed graphs. To the best
of our knowledge this is the first significant improvement on the bound for
Younger's conjecture since it was proved in 1996.
  We believe that the theoretical tools we developed may find applications
beyond the directed grid theorem, in a similar way as the path-of-sets-system
framework due to Chekuri and Chuzhoy (2016) did (see for example Hatzel,
Komosa, Pilipczuk and Sorge (2022); Chekuri and Chuzhoy (2015); Chuzhoy and
Nimavat (2019)).
