How quickly can you get your child into his/her doctor when they are sick? that day or the next? PLEASE ANSWER?

my newborn baby girl has been to her first checkup already but it took 2 weeks to get an appt (and that was only cuz there was a cancelation) AND i called to set it up the day before she was born (i was induced, so i knew it was time lol). and now here it is a week later and i believe she is having some upper respitory problems. she is congested, the nurse said use sailine drops and booger sucker. well its not working the greatest and now snot is kinda bloody. her eyes are also watering. i asked for an appt and they dont have anything for 4 weeks!!!! they just told me if i was worried about it, that i should take her to childrens hospital (WHICH IS PRETTY FAR drive for me and in the GHETTO at that!!) not that i wont take her there cuz i have to..but i think its BS that i cant get her in there for an appt. shouldnt i be able to get her in alot sooner than 4 weeks?? (which is past her 2 month appt that i already set). it seems like when i was a kid, my mom could call me in an appt that day if i was sick. how quickly can you get your kids into the doc? or does everyone have to go the hospital nowadays?? im thinking i should switch docs for her but not sure if i will just have the same problem. thanks for your time if you help me out!!