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Icebergs calving into Greenlandic Fjords frequently experience strongly sheared flows over their draft, but the impact of this flow past the iceberg is not fully captured by existing parameterizations. We present a series of novel laboratory experiments to determine the dependence of submarine melting along iceberg sides on a background flow. We show, for the first time, that two distinct regimes of melting exist depending on the flow magnitude and consequent behavior of melt plumes (side-attached or side-detached), with correspondingly different meltwater spreading characteristics. When this velocity dependence is included in melt parameterizations, melt rates estimated for observed icebergs in the attached regime increase, consistent with observed iceberg submarine melt rates. We show that both attached and detached plume regimes are relevant to icebergs observed in a Greenland fjord. Further, depending on the regime, iceberg meltwater may either be confined to a surface layer or distributed over the iceberg draft.