As a new achievability scheme, successive refinement coding based compress-and-forward relaying (sCF) is proposed for both discrete memoryless and Gaussian broadcast relay channels (BRC) with one source, two destinations and a dedicated relay. A new outer bound is also provided on the capacity region of physically degraded BRC. Then, the achievability scheme is extended to K-destinations. For the Gaussian BRC, it is shown that the sCF rate region outperforms other CF schemes in the literature. Finally, it is proved that the new achievable scheme is within a half bit of the outer bound on the capacity region for a subset of channel gains.