Bipartite graphs are rich data structures with prevalent applications and identifier structural features. However, less is known about their growth patterns, particularly in streaming settings. Current works study the patterns of static or aggregated temporal graphs optimized for certain down-stream analytics or ignoring multipartite/non-stationary data distributions, emergence patterns of subgraphs, and streaming paradigms. To address these, we perform statistical network analysis over web log streams and identify the governing patterns underlying the bursty emergence of mesoscopic building blocks, 2,2-bicliques known as butterflies, leading to a phenomenon that we call "scale-invariant strength assortativity of streaming butterflies". We provide the graph-theoretic explanation of this phenomenon. We further introduce a set of micro-mechanics in the body of a streaming growth algorithm, sGrow, to pinpoint the generative origins. sGrow supports streaming paradigms, emergence of 4-vertex graphlets, and provides user-specified configurations for the scale, burstiness, level of strength assortativity, probability of out-of-order records, generation time, and time-sensitive connections. Comprehensive Evaluations on pattern reproducing and stress testing validate the effectiveness, efficiency, and robustness of sGrow in realization of the observed patterns independent of initial conditions, scale, temporal characteristics, and model configurations. Theoretical and experimental analysis verify the robust ability of sGrow in generating streaming graphs based on user-specified configurations that affect the scale and burstiness of the stream, level of strength assortativity, probability of-of-order streaming records, generation time, and time-sensitive connections.