We consider large uniform labeled random graphs in different classes with few induced $P_4$ ($P_4$ is the graph consisting of a single line of $4$ vertices), which generalize the case of cographs. Our main result is the convergence to a Brownian limit object in the space of graphons. We also obtain an equivalent of the number of graphs of size $n$ in the different classes. Finally we estimate the expected number of induced graphs isomorphic to a fixed graph $H$ for a large variety of graphs $H$. Our proofs rely on tree encoding of graphs. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.