We consider critical oriented Bernoulli percolation on the square lattice $\mathbb{Z}^2$. We prove a Russo-Seymour-Welsh type result which allows us to derive several new results concerning the critical behavior: - We establish that the probability that the origin is connected to distance $n$ decays polynomially fast in $n$. - We prove that the critical cluster of the origin conditioned to survive to distance $n$ has a typical width $w_n$ satisfying $εn^{2/5} < w_n < n^{1-ε}$ for some $ε> 0$. The sub-linear polynomial fluctuations contrast with the supercritical regime where $w_n$ is known to behave linearly in $n$. It is also different from the critical picture obtained for non-oriented Bernoulli percolation, in which the scaling limit is non-degenerate in both directions. All our results extend to the graphical representation of the one-dimensional contact process.