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This paper improves the sensitivity of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mi>ρ</mi></msup></semantics></math></inline-formula> family of weighted log-rank tests for the accelerated failure time model, accommodating realistic alternatives in survival analysis with censored data, such as heavy censoring and crossing hazards. The procedures are based on a weight function with the censoring proportion incorporated as a component. Extensive simulations show that the weight function enhances the performance of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>G</mi><mi>ρ</mi></msup></semantics></math></inline-formula> family, increasing its sensitivity and flexibility. The weight function method is illustrated with an example concerning vaginal cancer.