The lit portion of the moon always appears to "point" to the sun. When the moon is a crescent, it is frequently not bright enough to be seen unless the sun is below the horizon, but its apparent position relative to the sun is also very close, so a waxing crescent shortly after the new moon is only visible just after sunset, near the horizon in the west, where it sets very shortly thereafter. Conversely, a waning crescent in the days before a new moon rises in the east just before sunrise and is soon impossible to see because the sunlight overwhelms it.
In both cases, the crescent will appear as a U for an observer on the equator. Because the convex edge of a crescent moon always points toward the sun, and because the moon is easiest to see when the sun is below the horizon, crescent moons are most frequently seen with their convex edge closest to the horizon. However, wider crescents, closer to half full, are visible in broad daylight, so you may see a nearly half-full moon rising in the east shortly before solar noon (or setting in the west shortly after) with its concave edge pointing down.
If you could somehow see the 1-day-old crescent moon as if the sun were not there, you would see it rise shortly after the sun with the illuminated portion at the top of the disk. It would then follow the sun through the sky directly overhead (at the equinox, at least) with the illuminated portion of the disk always closest to the sun, and it would set shortly after the sun with the illuminated portion of the disk at the bottom.
If this seems counterintuitive, remember that it isn't the moon that has turned around but you. When you observe the sunrise or moonrise, you're facing to the east, and when you observe the sunset or moonset, you're facing to the west.