Linear Regression Assumption

the following website states that linear regression assumes a linear relationship between dependent and independent variables: "First, linear regression needs the relationship between the independent and dependent variables to be linear. " from: https://www.statisticssolutions.com/assumptions-of-linear-regression/

However, I have read elsewhere that linear regression can be applied to non-linear relationships, such as fitting a curved line such as f(x) = x^2 to a set of data.

Which one is correct?

Both are correct. You still will have a linear relationship: you just think of $$x_i^2$$ as a variable itself, so there is a linear relationship between $$y_i$$ and the variable $$x_i^2$$.
Linearity in linear regression is a linearity w.r.t. the coefficients, $$\beta$$. Formally, a model called linear if its gradient w.r.t. $$\beta$$ is independents of $$\beta$$, namely, let $$y=\beta_0+\sum_{j=1}^p\beta_jg_j(x_j)+\epsilon,$$ and $$\nabla_{\beta}\mathbb{E}[y|x_1,...,x_p]=(1, g_1(x_1),....,g_2(x_2))^T,$$ where $$g_j(x_j)$$ can be whatever you like, i.e., $$x_j^2, \sin x_j$$, etc., as long as they don't involve $$\beta$$s.