Soft exosuits used for supporting human muscle strength must be lightweight and wearable. Shape memory alloy (SMA) spring-based fabric muscles (SFM) are light and flexible, making them suitable for soft and shape-conformable exosuits. However, SFMs have a slow actuation speed owing to the slow cooling rate of the SMA spring. This paper proposes a forced air-cooling fan-integrated fabric muscle (FCFM) that improves the cooling rate by arranging a thin-diameter SMA spring bundle with a high surface-area-to-volume ratio inside a breathable fabric with integrated fans. The relaxation time of an FCFM weighing 30 g and containing a 2.6 g SMA spring bundle, which contains 200 thin springs, was reduced by over 70.2% via forced-air cooling using the integrated fans. A 4 kg weight, which is 1530 times the mass of the SMA spring bundle, was hung from the FCFM and was repeatedly actuated in ten-second cycles. An upper limb assistive soft exosuit with FCFMs was fabricated and worn on a mannequin holding a dumbbell, and the arm extension time after flexion was improved by 4.5 times. Additionally, the assistive performance of the exosuits for repetitive tasks in specific scenarios was evaluated, and the strong potential of the proposed FCFM for soft exosuits was verified.