Metrizability vs. Fréchet-Uryshon property

In metrizable spaces, points in the closure of a subset $A$ are limits of sequences in $A$; i.e., metrizable spaces are Fréchet-Uryshon spaces. The aim of this paper is to prove that metrizability and the Fréchet-Uryshon property are actually equivalent for a large class of locally convex spaces that includes $(LF)$- and $(DF)$-spaces. We introduce and study countable bounded tightness of a topological space, a property which implies countable tightness and is strictly weaker than the Fréchet-Urysohn property. We provide applications of our results to, for instance, the space of distributions $\mathfrak{D}'(\Omega)$. The space $\mathfrak{D}'(\Omega)$ is not Fréchet-Urysohn, has countable tightness, but its bounded tightness is uncountable. The results properly extend previous work in this direction.