While there are benefits of wide exposure and experience that comes from working on different projects, changing teams and projects every quarter can be counter-productive to one’s career growth. Learning and development is a time consuming process and typically, it takes a PM or Engineer 6-8 weeks to ramp up on new teams and projects. In other words, 12 weeks is usually not enough time to deliver significant success (in a repeatable way) on any medium to large scale project. In addition to that, if you’re changing teams every quarter, then there is a lot of context change involved which can negatively impact one’s productivity and ability to deliver fast results. If I were in this position, I would seek leadership support to switch to a more stable role internally, but if that doesn’t work then I would switch companies. Here are some of my suggestions to address this chaotic situation -

1. Open up to your Leadership: Open up to your leadership and explain to them how the re-orgs are impacting the morale and productivity of you and your team members. An empathetic leader will genuinely work on improving your situation.
2. Expand your value to the company: Ask yourself how you can add more value to the company or the projects that you’re working on. How can you positively expand the scope of your projects and demonstrate the impact of your initiatives in a short span of time. The more value you can demonstrate, the higher will be your leadership’s motivation to keep you on the same team/project.
3. Improve communication and visibility: Reorganizations and chaos can negatively impact anyone's ability to effectively communicate with your stakeholders. However, in these situations, it is even more important to increase your visibility and improve communication with your leadership and peers. Try to use every opportunity to engage with your leadership and provide them necessary product updates or seek critical feedback from them on product and strategy.  

In short, quarterly reorganizations are counterproductive to your career growth and one can take the following measures to address such a chaotic situation - 1/ open up to your leadership and share your concerns, 2/ expand your value to the company and 3/ improve your communication and visibility in the company. However, if these measures don't yield results, despite your best efforts, then I would try to switch to a different company that can provide you with a better path for your career growth.

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