A strong edge coloring of a graph is a proper edge coloring such that no edge has two incident edges of the same color. Erdős and Nešetřil conjectured in 1989 that $5 /4∆2$ colors are always enough for a strong edge coloring, where $∆$ is the maximum degree of the graph. In the specific case where $∆ = 4$, we prove this to be true when there is no subgraph with average degree at least $4 − 1/5$ , and show that fewer colors are necessary when the graph is even sparser.