We investigate the magnetic phase diagram of a solid solution BaFe_{2}(S_{1-x}Se_{x})_{3}, which is categorized into a new family of Fe-based superconductors hosting a quasi-one-dimensional ladder lattice. We show that the antiferromagnetic order is largely suppressed due to bicritical fluctuations in the vicinity of the magnetic transition at x=0.23 from the stripe-type to block-type ordering. Moreover, we observed large magnon contributions in the thermal conductance in a wide x range. These results indicate the quasi-one-dimensional nature of spin systems and give a solid foundation for modeling the material toward unraveling the microscopic mechanism of superconductivity.