Most teams reach for sharding years before they need it. There is a ladder of cheaper, lower-risk fixes that comes first, and the right rung depends on which bottleneck you actually have: read load, write load, or data size. Here is the ladder, from cheapest to last resort.
Both are excellent databases and for most SaaS workloads either one is fine. The honest answer: pick Postgres for greenfield, keep what you run well, and switch only for a reason you can name. Here are the few differences that actually decide it.
Most slow-query problems are not a missing index, they are the wrong index, or ten indexes nobody needs. Start from the query plan, build the index the query actually wants, and drop the ones taxing every write.
RDS is often the second or third biggest line on an AWS bill, sometimes the first, and most of it is avoidable. The levers that move it: fix the queries before you upsize, match the instance to your load shape, and stop provisioning storage for data that has not arrived yet.
