You were correct about the use of proxies. They will make editing much easier. Combine proxies with Preview Scaling for best results.
Click to select a video clip on the timeline, go to the Properties panel, drop down the hamburger menu, and look for Convert to Edit-Friendly. In future projects, it may be easier to bring footage into the Source player first, convert it there, then use the converted video on the timeline from the beginning.
The Convert to Edit-Friendly box will provide three quality options… high-quality lossy, visually lossless intermediate, and mathematically lossless. Given that the sources are cell phones rather than cinema cameras, the high-quality lossy is probably a great option in terms of lower disk usage and time to convert and no noticeable loss in quality.