Over the past few years, a number of private networks have emerged. In these private networks, a server provides its subscribed clients with Internet services, forming a one-to-many network topology. Given the fact that users are located at different distances from the server, usage of the Transmission Control Protocol (TCP) for communication results in drastically unfair bandwidth allocations among the users. In this regard, this paper addresses the fairness and efficiency issues of TCP in such one-to-many IP (Internet Protocol) networks. The efficiency of TCP is controlled by matching the aggregate traffic rate of all TCP connections to the sum of the link capacity and total buffer size. On the other hand, its unfairness issue is mitigated by allocating bandwidth among individual flows in relative proportion with their RTTs. Simulation results elucidate that the proposed method makes better utilization of the network resources, reduces the number of packet drops, and provides a fair service to users.