Famous solvers like sedumi, sdpt3, mosek can solve conic optimization, but not more general convex optimization. Why? I know many convex problems can be formulated as conic, but still confused.
The fundamental algorithm employed by Mosek and SeDuMi is a primal-dual interior-point algorithm based on the work of Nesterov and Todd (NT). See for instance my paper and the references therein. This algorithm is only applicable to conic optimization problems over the symmetric cones. Therefore, it is not at all obvious or natural to extend those optimizers to general convex optimization. You can even argue theoretically that it is impossible to generalize it. Nevertheless in a recent paper we present a primal-dual interior-point algorithm for some nonsymmetric cones which is quite similar to the NT algorithm and seems to work well in practice.
Therefore, Mosek can currently deal with 5 cone types i.e. the linear, the quadratic, the semi-definite, the power and the exponential cone. It turns out that if you can deal with those 5 cone types you can handle almost all convex optimization that appears in practice. See the Mosek modeling cookbook for details. Also one of the more important exceptions is discussed at the Mosek blog.
Hence, there is a little value in building an optimizer for general convex optimization because it will have little use. Moreover, a conic optimizer specialized towards a small set of the most important cones will be easier to use and more efficient than some general purpose software. At least that is the bet we have made at Mosek.
PS. Up until version 8 MOSEK was capable of solving general convex optimization problems using the algorithm presented in yet another paper. However, it was rarely used and often users tried to apply it to nonconvex problems causing support issues.