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This letter reports experimental observations of the flow kinematics and stability of thin fluid sheets produced by impinging obliquely aligned laminar jets of a series of viscoelastic worm-like micelle solutions. As the velocity of the impinging jets is increased, the sheets of viscoelastic fluid grow larger and eventually become unstable. High speed imaging reveals a transition to a striking new flow structure resembling fluid webs, previously unobserved for Newtonian or non-Newtonian fluids. These newly discovered flow structures are complex and comprised of highly interconnected filaments created by the growth of multiple internal failures within the fluid sheet. Increasing viscoelasticity of the test fluid was found to stabilize the sheets and the fluid webs while increasing the drop size produced by their eventual breakup under capillary stresses.