I am sorry but I don't understand your question. We need context and sentences.
Both 'bulging' and 'popping out' suggest there's something wrong with the tyre. If the tyre had been flat we would say it's now 'inflated' and everybody would understand what is meant. It's full of air evenly distributed throughout the tyre.
'Bulging' suggests that one part is more full of air than the rest.
I can't think that 'pops out' is what you are looking for. It sounds more like a term for a puncture but then we say the tyre 'blew'. This just might be very American English.