Bitcoin's attempted solution on this front is off-chain scaling via lightning network. As far as I can ascertain, this has had highly limited adoption.

Eth's attempted solution on this front is sharding. I can't claim to be an expert in this, but from my understanding after proof-of-stake is deployed, ethereum plans to deploy something like 64 separate "shards" which, from my understanding, are like extra blockchains for conducting transactions, and using some kind of complicated proof of stake system to keep it consistent. In this case, while the main-net still has limited global throughput, scaling up to add more side chains will allow scaling additional throughput. You can read more here https://ethereum.org/en/eth2/

As with lightning, we don't know how well this will actually end up going until it's deployed.