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Why do I need Bridge mode on my main router to connect a second one?

I want to buy a better router than the one my ISP provides me. I can’t just ditch my ISP router because it is modem and router at the same time.

What I’ve been told is that I would need to set my ISP router in bridge mode to have a second one managing all my home network.

My ISP router doesn’t have a bridge mode option. I’ve been told that I could use a DMZ configuration and it would be the same.

My true question is why do I need my first router with bridge mode in the first place?

Example: I connected a old router I have to the ISP router. LAN port on my ISP router, WAN port on the “old one”. Addresses of 192.168.1.x in my ISP router, 192.168.0.x on the “old one”. Everything seemed to work fine on both routers. I was told that with this configuration I would have no internet access in the “old one”, but I had.