How can I improve my team’s process for testing our web-app?

Could you possibly clarify? This is a bit broad: like, what kind of testing do you currently have in place (if any)? You didn't really describe your process.

There are a lot of different kinds of tests you could do: unit tests for code coverage, E2E testing via automated tests through Playwright or Cypress, visual regression testing through a SaaS service that take screenshots of your webpage, cross-browser testing, integration testing between services, etc.

All these are forms of tests for web applications. So, it is difficult to suggest specifics to you without knowing if you are doing any of this testing already.

Playwright