Let \(q_0 = 2^s\) with \(s\geq 1\) and \(q = 2q^2_0\). Let \(G\) be a connected reductive algebraic group of type \(^2B_2\) over \(K=\mathbb F_q\) and let \(S\) be the Deligne-Lusztig variety associated to \(G\). One knows that \(S\) is an irreducible curve of genus \(q_0(q-1)\) with \(q^2+1\) points over \(K\), and its automorphism group is the Suzuki group \(Sz(q)\) of order \(q^2(q-1)(q^2+1)\). The main purpose of the paper is to investigate the numbers of \(K\)-rational points of the quotient curves of \(S\) by various subgroups \(H\) of \(Sz(q)\). In particular when \(H\) is a cyclic subgroup of order \(r|q-1\), and when \(H\) is a subgroup of the Singer subgroups of \(S7(q)\), the authors give their concrete plane models and determine their numbers of \(K\)-rational points. The numbers are in the interval given in the tables of curves with many rational points in [G. van der Geer and M. van der Vlugt, see http://www.wins.uva.nl/\(\sim\)geer].