Let $X$ be a rearrangement-invariant space. An operator $T:
X\to X$ is called narrow if for each measurable set $A$ and
each $\varepsilon > 0$ there exists $x \in X$ with $x^2=
\chi_A,\ \int x d \mu = 0$ and $\| Tx \| < \varepsilon$. In
particular, all compact operators are narrow. We prove that if
$X$ is a Lorentz function space $L_{w,p}$ on [0,1] with
$p > 2$, then there exists a constant $k_X > 1$ such that
for every narrow projection $P$ on $L_{w,p}$ $\|
\operatorname{Id} - P \| \geq k_X. $ This generalizes earlier
results on $L_p$ and partially answers a question of
E. M. Semenov. Moreover, we prove that every
rearrangement-invariant function space $X$ with an absolutely
continuous norm contains a complemented subspace isomorphic to
$X$ which is the range of a narrow projection and a non-narrow
projection. This gives a negative answer to a question of
A. Plichko and M. Popov.