Isbell shows in *[Function spaces and adjoints][1]* (1975) that every cocontinuous functor $\mathbf{Top} \to \mathbf{Set}$ is a coproduct of copies of the forgetful functor. (I have found an alternative proof for this within a more general theory, details will be added here when it's online.) But a necessary condition for a category to be modelled by a (possibly large) colimit sketch is  that the cocontinuous functors to $\mathbf{Set}$ are jointly conservative. By picking any bijective continuous map which is not an isomorphism, we get a contradiction.

*Edit. An alternative proof can be found in my new paper [*Large limit sketches and topological space objects*](https://arxiv.org/abs/2106.11115) (2021), Theorem 8.7.

  [1]: https://www.mscand.dk/article/view/11581/9597