Database Change Management

What are ways in which teams handle database change management? I have a team of 8 developers making database changes concurrently. Each developer is responsible for updating their changes in what we consider a "poor man's database change log" which is essentially a shared file referencing a script. Sometimes the scripts are database alters. Other times the script re-creates a view due to a change (which may have also been changed by another developer later on). It always happens that scripts are placed out-of-order, listed in duplicate, etc. My gut tells me we are doing this wrong. How can we do this better? If it matters, we are using Postgres + git.