Despite growing adoption of cryptocurrencies, making fast payments at scale remains a challenge. Payment channel networks (PCNs) such as the Lightning Network have emerged as a viable scaling solution. However, completing payments on PCNs is challenging: payments must be routed on paths with sufficient funds. As payments flow over a single channel (link) in the same direction, the channel eventually becomes depleted and cannot support further payments in that direction; hence, naive routing schemes like shortest-path routing can deplete key payment channels and paralyze the system. Today's PCNs also route payments atomically, worsening the problem. In this paper, we present Spider, a routing solution that “packetizes” transactions and uses a multi-path transport protocol to achieve high-throughput routing in PCNs. Packetization allows Spider to complete even large transactions on low-capacity payment channels over time, while the multi-path congestion control protocol ensures balanced utilization of channels and fairness across flows. Extensive simulations comparing Spider with state-of-the-art approaches shows that Spider requires less than 25% of the funds to successfully route over 95% of transactions on balanced traffic demands, and offloads 4x more transactions onto the PCN on imbalanced demands.