Ignore comparing absolute bandwidth numbers between the machines. The old AMD GPUs in the PS4 and the PS4 Pro are much more bandwidth hungry than modern GPUs. It's not apples to apples. What you actually want to look at is "bandwidth per TFLOP of compute" and "is this enough bandwidth for the GPU to stay fed."
RTX 30 GPUs all sit at 25-30 GB/s/TFLOP, and the ones at the top of that range perform more consistently than the ones at the bottom (see: shitloads of trawling through Digital Foundry benchmarks).
Consoles also share bandwidth between the CPU and the GPU, and unlike the GPU, T239 will likely use the same clock for the CPU in both modes (because game logic doesn’t scale with resolution).
4 TFLOPS (the PS4 Pro’s number) while docked brings you to 25.5 GB/s/TFLOP, in the underperforming range even before you take out a premium for the CPU.
Handheld is, presumably at 2/3rds the bandwidth (using standard LPDDR clocks). The PS4 is only 1.8 TFLOPS. With less than half the GPU power, and 2/3rds of the bandwidth, we’re obviously in better shape. 37.7 GB/s/TFLOP is well past the top RTX 30 numbers, while leaving plenty of room for the CPU.
So, pushing PS4 level of TFLOPs while keeping the GPU and the CPU fed isn’t a problem in either mode. But past ~3TFLOPS, bandwidth becomes an issue.
Appendix: Why is bandwidth so much lower on modern GPUs?
The GCN GPU used in the PS4 and Xbone was very bandwidth hungry. Credit to
@Pokemaniac for explaining this to me. Basically it renders a screen at a time, front to back. For SD resolutions that’s not awful, but on HD or higher screens this is an aggressively cache-unfriendly design. You’re having to fetch the same data over and over again as you retouch it.
Later GPUs, including the one in the Switch, cut the screen into tiles, and can render multiple tiles at a time. The tile size is dynamically selected to fit into cache, so data only needs to be loaded once, and parallel rendering of tiles allows the small amount of bandwidth to be used constantly, instead of bursty load-render-load-render like older GPUs.