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To make statistical inferences about regression model parameters, it is necessary to assume a specific distribution on the random error expression. A basic assumption in a linear regression model is that the random error expression follows a normal distribution. However, in some statistical researches, data simultaneously display skewness and bimodality features. In this setting, the normality assumption is  violated. A common approach to avoiding this problem is to use a mixture of skew-normal distributions. But such models involve many parameters, which it makes difficult to fit the models to the data. Moreover, these models are faced with the non-identifiability issue.
In this situation, a suitable solution is to use flexible distributions, which can take into account the skewness and bimodality observed in the data distribution. In this direction, various methods have been proposed based on developing of the skew-normal distribution in recent years. In this paper, these methods are used to introduce more flexible regression models than the regression models based on skew-normal distribution and a mixture of two skew-normal distributions. Their performance is compared using a simulation example. The methodology is then illustrated in a practical example related to a horses dataset.