The production of broadband, terawatt terahertz (THz) pulses has been demonstrated by irradiating relativistic lasers on solid targets. However, the generation of extremely powerful, narrow-band and frequency-tunable THz pulses remains a challenge. Here, we present a novel approach for such THz pulses, in which a plasma wiggler is elaborated by a tabletop laser and a near-critical density plasma. In such a wiggler, the laser-accelerated electrons emit THz radiations with a period closely related to the plasma thickness. The theoretical model and numerical simulations predict that a THz pulse with a laser-THz energy conversion of over 2.0%, an ultra-strong field exceeding 80 GV/m, a divergence angle of approximately 20 • and a center frequency tunable from 4.4 to 1.5 THz can be generated from a laser of 430 mJ. Furthermore, we demonstrate that this method can work across a wide range of laser and plasma parameters, offering potential for future applications with extremely powerful THz pulses.