no, when you create a character you will first level to 10, or start at 10 if it’s an allied race toon. then at lvl 10 you’ll be able to choose to go play through the burning crusade, or wrath, or cataclysm, or pandaria, or WoD, or Legion, or BfA, to go from 10-50. each expansion will have enough quests and content to get you from 10-50. then the shadowlands content will begin and you go from 50-60.
there will be no “60-120”, not for anything. 60 will be the new level cap. and probably for every expansion that comes after shadowlands as well, resetting max-level toons back to 50 for the beginning of each new xpac and dropping the previous one down to a levelling xpac.