  In this paper we investigate the existence of subexponential parameterized
algorithms of three fundamental cycle-hitting problems in geometric graph
classes. The considered problems, \textsc{Triangle Hitting} (TH),
\textsc{Feedback Vertex Set} (FVS), and \textsc{Odd Cycle Transversal} (OCT)
ask for the existence in a graph $G$ of a set $X$ of at most $k$ vertices such
that $G-X$ is, respectively, triangle-free, acyclic, or bipartite. Such
subexponential parameterized algorithms are known to exist in planar and even
$H$-minor free graphs from bidimensionality theory [Demaine et al., JACM 2005],
and there is a recent line of work lifting these results to geometric graph
classes consisting of intersection of "fat" objects ([Grigoriev et al., FOCS
2022] and [Lokshtanov et al., SODA 2022]). In this paper we focus on "thin"
objects by considering intersection graphs of segments in the plane with $d$
possible slopes ($d$-DIR graphs) and contact graphs of segments in the plane.
Assuming the ETH, we rule out the existence of algorithms:
  - solving TH in time $2^{o(n)}$ in 2-DIR graphs; and
  - solving TH, FVS, and OCT in time $2^{o(\sqrt{n})}$ in $K_{2,2}$-free
contact 2-DIR graphs.
  These results indicate that additional restrictions are necessary in order to
obtain subexponential parameterized algorithms for %these problems. In this
direction we provide:
  - a $2^{O(k^{3/4}\cdot \log k)}n^{O(1)}$-time algorithm for FVS in contact
segment graphs;
  - a $2^{O(\sqrt d\cdot t^2 \log t\cdot k^{2/3}\log k)} n^{O(1)}$-time
algorithm for TH in $K_{t,t}$-free $d$-DIR graphs; and
  - a $2^{O(k^{7/9}\log^{3/2}k)} n^{O(1)}$-time algorithm for TH in contact
segment graphs.
