Twin-width is a structural width parameter introduced by Bonnet, Kim,
Thomass\'e and Watrigant [FOCS 2020]. Very briefly, its essence is a gradual
reduction (a contraction sequence) of the given graph down to a single vertex
while maintaining limited difference of neighbourhoods of the vertices, and it
can be seen as widely generalizing several other traditional structural
parameters. Having such a sequence at hand allows to solve many otherwise hard
problems efficiently. Our paper focuses on a comparison of twin-width to the
more traditional tree-width on sparse graphs. Namely, we prove that if a graph
G of twin-width at most 2 contains no Kt,t subgraph for some integer
t, then the tree-width of G is bounded by a polynomial function of t. As
a consequence, for any sparse graph class C we obtain a polynomial
time algorithm which for any input graph G∈C either outputs a
contraction sequence of width at most c (where c depends only on
C), or correctly outputs that G has twin-width more than 2. On
the other hand, we present an easy example of a graph class of twin-width 3
with unbounded tree-width, showing that our result cannot be extended to higher
values of twin-width