It is called a waxing gibbous
A moon that is larger than a quarter can be referred to as a "large moon" or simply a "moon." In astronomical terms, moons are typically classified based on their size, with larger moons often being called "large satellites." However, there isn't a specific term that universally defines moons based solely on their size relative to a quarter.
Because it is the closest star to us. This is the same reason that a quarter looks larger than the moon if you hold it up. The quarter is much closer to you than the moon, so it looks bigger, but it is not really bigger than the moon.
No moon is larger than the planet it orbits. The two largest moons in the solar system, Ganymede and Titan, which orbit Jupiter and Saturn respectively, are larger than the planet Mercury, but no others.
That's the "gibbous" phase.
The moon phase between waxing crescent and first quarter is called waxing gibbous. In this phase, the moon is more than half illuminated but less than full.
When the moon is becoming full, it is called the waxing gibbous phase. It occurs after the first quarter moon and before the full moon, when more than half but not all of the moon's face is illuminated.
As the moon transitions from the new moon phase to the first quarter phase, the visible portion of its surface, or the illuminated part, increases. This is due to the moon's orbit around the Earth, which allows more sunlight to illuminate its surface. During this time, the crescent shape of the moon gradually expands, leading to a larger visible portion. Eventually, it will reach the first quarter phase, where half of the moon will be illuminated.
When more than half of the moon is illuminated as seen from Earth, it is called a "gibbous" moon. Specifically, if it is waxing (growing toward a full moon), it is referred to as a "waxing gibbous," while if it is waning (decreasing after a full moon), it is called a "waning gibbous." This phase occurs between the first quarter and full moon, and again between the full moon and last quarter.
The moon is unusually large in proportion to Earth at just over a quarter its diameter. While several moons in the solar system are larger than ours, they orbit much larger planets.
It is called a crescent, and if it is growing, it's called a waxing crescent, otherwise, it's a waning crescent. So, when less than half is facing the earth, it is called a Waning Cresent
The moon phase you are describing is the waxing crescent phase. During this phase, less than half of the moon is illuminated, and the visible portion is increasing as it progresses toward the first quarter phase. The illumination grows as the moon orbits the Earth, moving away from the new moon phase.
The lessening of the illumninated part of the Moon is called "waning" (decreasing). The phases included are waning gibbous (more than 50%), last quarter, waning crescent (less than 50%), and finally the new moon.