to bear the weight of something: to carry, lift, or hold it up
At its core this verb is about taking the full weight of a thing onto yourself. A person carries a jar of water, lifts a sick man to the temple gate, or shoulders a cross. From that plain physical picture the word stretches naturally in several directions, but the picture never disappears: whatever is being carried has real weight, and someone is holding it up.
The most important stretch is from carrying a load to bearing a burden that is not physical at all. To carry someone's failings, to put up with difficult people, or to endure a hard demand is to take their weight the same way you would take a sack onto your back. This is why the letters can say "carry one another's burdens" and also "each will carry his own load" in the same breath: one is the weight we help others hold, the other is the weight only we can answer for.
The word also reaches toward lifting up (picking up stones to throw), toward carrying away (removing something, and once the shady sense of helping oneself to what belongs to others), and toward holding up from underneath (a root that supports a tree, a creature that carries a rider). English needs a whole family of words for all this, carry, bear, lift, take up, endure, remove, but in Greek it is one verb with one idea at the center.