We investigate the O($n$) nonintersecting loop model on the square lattice
under the constraint that the loops consist of ninety-degree bends only. The
model is governed by the loop weight $n$, a weight $x$ for each vertex of the
lattice visited once by a loop, and a weight $z$ for each vertex visited twice
by a loop. We explore the $(x,z)$ phase diagram for some values of $n$. For
$0<n<1$, the diagram has the same topology as the generic O($n$) phase diagram
with $n<2$, with a first-order line when $z$ starts to dominate, and an
O($n$)-like transition when $x$ starts to dominate. Both lines meet in an
exactly solved higher critical point. For $n>1$, the O($n$)-like transition
line appears to be absent. Thus, for $z=0$, the $(n,x)$ phase diagram displays
a line of phase transitions for $n\le 1$. The line ends at $n=1$ in an
infinite-order transition. We determine the conformal anomaly and the critical
exponents along this line. These results agree accurately with a recent
proposal for the universal classification of this type of model, at least in
most of the range $-1 \leq n \leq 1$. We also determine the exponent describing
crossover to the generic O($n$) universality class, by introducing topological
defects associated with the introduction of `straight' vertices violating the
ninety-degree-bend rule. These results are obtained by means of transfer-matrix
calculations and finite-size scaling.Comment: 19 pages, 11 figure