Motivated by recent discovery of yttrium-based high-temperature ternary superconducting hydrides (e.g., CaYH$_{12}$, LaYH$_{12}$, and ScYH$_{6}$), we have employed evolutionary algorithm and first-principles calculations to comprehensively examine the structural stability and superconductivity of the YMgH$_{x}$ system at high pressure. The hydrogen content $x$ and the pressure are both important factors in the stability of these candidate structures. We find that the stability of hydrogen-rich materials frequently necessitates higher pressure. For instance, the pressures to stabilize $P4/mmm$-YMgH$_{8}$ and $Cmmm$-YMgH$_{12}$ are both more than 250 GPa. Hydrogen-less materials, such as $I4_{1}/amd$-YMgH$_{2}$ and $P6_{3}/mmc$-YMgH$_{3}$, can be stable at pressures as low as 100 GPa. In addition, we find a metastable structure for YMgH$_{6}$ with the same space group as the $P4/mmm$-YMgH$_{8}$. A metastable sodalite-like face-centered cubic (FCC) structure is also found in YMgH$_{12}$. These four clathrate structures of $P4/mmm$-YMgH$_{6}$, $P4/mmm$-YMgH$_{8}$, $Cmmm$-YMgH$_{12}$, and $Fdbar{3}m$-YMgH$_{12}$ is made up of H14, H18, H24, and H24 cages, respectively, in which the H-H pair exhibits weak covalent bonding. According to phonon calculations, $P4/mmm$-YMgH$_{6}$ and $P4/mmm$-YMgH$_{8}$ require a pressure of 300 GPa to maintain dynamic stability, however $Cmmm$-YMgH$_{12}$ and $Fdbar{3}m$-YMgH$_{12}$ can maintain dynamic stability at pressures of 200 GPa and 250 GPa, respectively. Electron-phonon coupling calculations indicate that they might be potential high-temperature superconductors, with superconductivity intimately linked to the H cage structure. The sodalite structure $Fdbar{3}m$-YMgH$_{12}$ has a $T_mathrm{c}$ value of 190 K and a strong electron-phonon coupling constant of 2.18.
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