Best circuit to play music in car, using ISO 10487
LM386 power amplifiers are just too puny to overcome high ambient noise level inside a car. The cracking you describe could have two causes: power amplifier clipping, or speaker overload. Considering the weak driver (LM386), amplifier clipping likely causes distortion, and limits power output.
You certainly need a more powerful amplifier. Some considerations for you before choosing a solution:
What net speaker resistance will the amplifier drive? (8 ohms? 4 ohms?...)
From what DC voltage source will the amplifier be powered? (12.6v? custom?)

Your open-ended sound-level spec ("not as high as I would like it to be", "high bass is one of my priorities") is too subjective. It is possible that a bridged amplifier powered from the 12.6v car source will satisfy, if it drives a 4-ohm speaker load.
 At maximum volume (where clipping begins), conversation with an adjacent companion will be very difficult. If you need more volume, a higher DC supply voltage than 12.6v would be required. I am assuming old-school Class AB linear amplifier technology, similar to the low-power LM386.