目眩く (めくるめく, mekurumeku) describes something so dazzling or overwhelming that it makes you feel dizzy. It is a poetic, literary expression rather than an everyday word.
In the song, 目眩く恋 means “a dizzying, all-consuming romance,” though the singer immediately denies that their relationship is even that kind of passionate love. That contrast makes the word especially memorable and intriguing.
ROMANCE is a song sung by a man who wants everything from an affair and is willing to promise nothing. He opens by telling his lover to drop her pride, then admits he has no plans to be her boyfriend: "彼氏になる気はなくて" (I have no intention of becoming your boyfriend). He still gets jealous and possessive, though, and he still wants to matter: "君だけの特別でいたい" (I want to be special only to you). There is already a main partner in the picture, too. "本命" is slang for your "real" or number-one partner, and he plans to keep her fooled.
The chorus calls it all a "じれったいロマンス" (frustrating romance) and a "身勝手なロマンス" (selfish romance), and that is honest of him. His logic is cheeky: a relationship with no future has no ending, "だって始まってもないから" (because it never even started). Then he shrugs, "これも愛の形でしょ?" (this is a kind of love too, right?). The bridge gets very explicit, and the second verse is the coldest part: he offers to comfort her on lonely nights, but only "僕の彼女にバレない程度に" (as long as my girlfriend doesn't find out). He is a cad, and the song knows it, which is what makes it fun to sing along to.