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.