Linear models are a linear combination of possibly non-linear regressors. The linearity is strictly in the parameters, not in whatever you're adding up.

A neural network can be pedantically referred to as a linear model of the form y = a + b*neural_network, for example. Here, y is a linear model (even though neural_network isn't).

A linear relationship between any transformation of the outcome and any transformation of the predictor variables — so the function is linear but the relationship between predictors and outcome can take on almost any shape.