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In this paper we investigate the existence of subexponential parameterizedalgorithms of three fundamental cycle-hitting problems in geometric graphclasses. 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 suchthat $G-X$ is, respectively, triangle-free, acyclic, or bipartite. Suchsubexponential 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 graphclasses consisting of intersection of "fat" objects ([Grigoriev et al., FOCS2022] 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 $K_{2,2}$-freecontact 2-DIR graphs.  These results indicate that additional restrictions are necessary in order toobtain subexponential parameterized algorithms for %these problems. In thisdirection we provide:  - a $2^{O(k^{3/4}\cdot \log k)}n^{O(1)}$-time algorithm for FVS in contactsegment graphs;  - a $2^{O(\sqrt d\cdot t^2 \log t\cdot k^{2/3}\log k)} n^{O(1)}$-timealgorithm 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 contactsegment graphs.

Subexponential algorithms in geometric graphs via the subquadratic grid minor property: the role of local radius