A novel two-step inverse-scattering technique is proposed for through-the-wall microwave imaging. The approach is based on a regularization scheme developed in the framework of variable-exponent Lebesgue spaces, which enhances the quality of the reconstruction by properly tuning the exponent function that defines the adopted norm. Such a function is built directly from the available data by using a beamforming technique based on a delay-and-sum scheme. After an initial numerical assessment, the approach is validated against experimental measurements in a laboratory environment, with targets placed behind a brick wall. Measured data are collected in time domain by scanning the transmitting and receiving antennas in a multi-illumination and multi-view arrangement. The processing of experimental data is performed in the frequency domain, where data at multiple frequencies are extracted by a Fast Fourier Transform (FFT) and simultaneously processed by the imaging algorithm. The obtained imaging results confirm the good reconstruction capabilities of the developed inverse-scattering scheme in the case of both metallic and low-contrast targets.