Generating intense ultrashort pulses with high-quality spatial modes is crucial for ultrafast and strong-field science and can be achieved by nonlinear supercontinuum generation (SCG) and pulse compression. In this work, we propose that the generation of quasi-stationary solitons in periodic layered Kerr media can greatly enhance the nonlinear lightmatter interaction and fundamentally improve the performance of SCG and pulse compression in condensed media. With both experimental and theoretical studies, we successfully identify these solitary modes and reveal their unified condition for stability. Space-time coupling is shown to strongly influence the stability of solitons, leading to variations in the spectral, spatial and temporal profiles of femtosecond pulses. Taking advantage of the unique characteristics of these solitary modes, we first demonstrate single-stage SCG and the compression of femtosecond pulses from 170 to 22 fs with an efficiency >85％. The high spatiotemporal quality of the compressed pulses is further confirmed by highharmonic generation. We also provide evidence of efficient mode self-cleaning, which suggests rich spatiotemporal self-organization of the laser beams in a nonlinear resonator. This work offers a route towards highly efficient, simple, stable and highly flexible SCG and pulse compression solutions for state-of-the-art ytterbium laser technology.