Although I can’t claim to know Lorde personally, it’s unlikely that she is transgender. Her level of fame is pretty high, if she were transgender it’s pretty likely the public would know by now!
Why would someone think Lorde was transgender?
When someone writes a rumour on the internet about anyone. celebrity or otherwise, it’s so easy for that rumour to spread like wildfire via sites like Twitter and gossip tabloids. Depending on how much of the stuff on the internet you trust, it’s easy to believe such stories.
Lorde's Style
Lorde has a quirky, sometimes androgynous clothing style in her music videos, photo shoots and so on. It can be difficult to put her in the same box as a lot of other female artists who dress more femininely, so maybe this contributes to some peoples curiosity about her gender.
South Park Episode
South Park, the adult cartoon show, ran an episode where they basically depicted Lorde as a 45 year old transgender man. It’s very likely that this is where the rumour about her being transgender originated from.