Confusion regarding sheaf cohomology and singular cohomolgy

Question

Let $X$ be a smooth, projective curve (in particular irreducible) of genus $g$ at least $1$. We know that $H^1_{\mbox{sing}}(X,\mathbb{Z})=2g$. But, since $X$ is irreducible, the locally constant sheaf $\mathbb{Z}$ is flasque, hence the sheaf cohomology $H^1(X,\mathbb{Z})=0$. My confusion lies in the fact that $X$ is locally contractible and it seems that the two cohomologies must coincide in this case. I do not understand, where I am going wrong. Can someone help!