It can illuminate an area of a darkness spell, as light from a magic item is magical light.
As per the darkness spell description, it blocks non-magical light:
A creature with darkvision can't see through this darkness, and nonmagical light can't illuminate it.
But does it explicitly allow magical light? Yes. Jeremy Crawford, whose rulings are considered official, says in two tweets:
Light from any magical source can illuminate the area of a darkness spell, but the darkness spell can dispel light created by a spell of 2nd level or lower, not light created by a non-spell.
If a source of magical light is not a spell of 2nd level or lower, darkness can be illuminated by that light.
Is light from a magic item "magical light"? According to our question What is considered magical light for the purposes of the Darkness spell?, light from a magic item is magical light.
I concur with this assessment. Words with no game rule meaning in D&D 5th edition (i.e. "magical light" isn't defined anywhere in the books) are interpreted by their natural English meaning. A magic item is clearly a "magical source", and I think it would be difficult to argue that light produced directly by a magic item is nonmagical.
Note that the level of a darkness spell doesn't have any effect, citing another Crawford tweet:
As written, the darkness spell can't be made more powerful with a higher level slot.
Therefore, Lightbringer, being magical, can illuminate an area affected by the darkness spell, regardless of level.
It can't actually dispel (i.e. end) the darkness spell, since no part of the item's description says it can. If you take Lightbringer back out of the area, for example, the darkness effect returns.