My request for the exchange of the product I purchased was turned down due to their reasoning that there is no conclusive evidence for the product's malfunction.
However, the product's malfunction occurred during its warranty period.
ご質問ありがとうございます。
"My request for the exchange of the product I purchased was"＝「私が購入した製品の交換要求は」
"turned down"＝「却下された」
"due to their reasoning that there is no conclusive evidence for the product's malfunction."＝「製品の不具合には明確な証拠がないという理由により。」
"However, the product's malfunction occurred during its warranty period."＝「しかし、製品の不具合は保証期間内に起こった。」
ご参考に。
アメリカ合衆国 
