bolster
To support something and make it stronger or more secure.
To bolster something means to strengthen it or provide extra support. When you bolster your argument in a debate, you add more evidence and examples to make it more convincing. When a coach bolsters her team's confidence before a big game, she builds them up with encouragement and reminders of their skills.
The word suggests propping something up that might otherwise be weak or wobbly. You might bolster a sagging bookshelf by adding an extra support beam underneath, or bolster your case for getting a pet by showing your parents how responsible you've been with your chores. A student might bolster their grade by doing extra credit work.
Originally, a bolster was a long pillow or cushion used to support your head or back, which is where we get this sense of providing support. When you bolster something, you're essentially putting a pillow under it, whether that's a physical object, an idea, someone's spirits, or your own position on an issue. The word carries a sense of making something stronger and more secure than it was before.