This paper presents a new class of sparse superposition codes for low-rates and short-packet communications over the additive white Gaussian noise channel. The new code is orthogonal sparse superposition (OSS) code. A codeword of OSS codes is represented as a superposition of sparse sub-codewords whose support sets are mutually non-overlapping. To construct such codewords in a computationally efficient manner, a successive encoding method is presented. Harnessing the orthogonal property among sub-codewords, a simple yet near-optimal decoding method is proposed, which performs element-wise maximum a posterior decoding with successive support set cancellation. This decoder is super-fast by a linear decoding complexity in block lengths, far less than the commercially used channel decoders for modern channel codes. The upper bounds for the block error rates (BLERs) are analytically derived for few-layered OSS codes as a function of block lengths and code rates. It turns out that a single-layered OSS code achieves the ultimate Shannon limit in the power-limited regime, even with the linear complexity decoder. Via simulations, the proposed OSS codes are shown to perform better than commercially used coded modulation techniques for low-rate and short-latency communication scenarios.