It has been reported that the IEEE 802.11 MAC protocol and the TCP congestion control are highly problematic in terms of flow starvation in wireless mesh networks (WMNs). However, the economic features of IEEE 802.11 make it the commonly-used MAC protocol in WMNs. Therefore, solving starvation at the transport layer seems to be more appropriate. Indeed, the main starvation cause in TCP is that congestion is managed as a link-based problem. However, since bandwidth is a spatially-shared resource in WMNs, congestion is a neighborhood phenomenon that should be handled using mutual cooperation within a congested neighborhood. Such cooperation considerably consumes the already scarce bandwidth of WMNs causing more congestion. In this paper, we propose a neighborhood-aware and overhead-free congestion control scheme (NICC) that solves the starvation problem without impacting the scarce bandwidth of WMNs. NICC makes use of some underexploited fields in the IEEE 802.11 frame header, without modifying the standard frame size, to provide an overhead-free multi-bit congestion feedback; being overhead-free, this feedback allows performing neighborhood cooperation without generating control overhead. Furthermore, being multi-bit, it yields source nodes a fine-grained indication of the congestion degree, providing accurate rate control. The NICC performance in terms of starvation avoidance and bandwidth efficiency is proven through extensive simulations.