Trouble with bridged TDA2050 and single supply.

If you are talking about the single supply TDA based version, it may be. The breadboard is not one of a best in the market but it's still a breadboard, we all know that they are not useful every time you need a good quality signal even in audio applications. However I moved every component to try making contact if there were missing ones and it was the same. By the way, the split supply version worked very well so I don't think it is a breadboard problem.

I designed a bridged TDA2030 amplifier and made the PCB, it sounded pretty good. I am going to make two separate bridges for the woofer and tweeter and cool it down with a PC cooler, it should keep the temperature below 100°C I think. Anyways I am planning to use a NTC to control the gain when the HS is near 120°C to avoid overheating. I know that the chip has a built in thermal turn off but I don't want it to stop sounding.

If the heatsink is 100 or 120 degrees C then the chip is much hotter, maybe too hot. The package for audio amplifier ICs is designed to be bolted to a finned cast aluminum heatsink with thermal compound grease in between. The size of the heatrsink can be calculated from its spec's and the thermal resistance of the IC.

The audio amplifier ICs do not have a ground pin so they do not know and do not care if the supply is single or split so both supplies should work the same.

An amplifier that is clipping produces less heat than one that is linear with a high output so reducing the volume when it gets too hot might be bad.