Doubly-selective channel estimation represents a key element in ensuring communication reliability in wireless systems. Due to the impact of multi-path propagation and Doppler interference in dynamic environments, doubly-selective channel estimation becomes challenging. Conventional channel estimation schemes encounter performance degradation in high mobility scenarios due to the usage of limited training pilots. Recently, deep learning (DL) has been utilized for doubly-selective channel estimation, where convolutional neural network (CNN) networks are employed in the frame-by-frame (FBF) channel estimation. However, CNN-based estimators require high complexity, making them impractical in real-case scenarios. For this purpose, we overcome this issue by proposing an optimized and robust bi-directional recurrent neural network (Bi-RNN) based channel estimator to accurately estimate the doubly-selective channel, especially in high mobility scenarios. The proposed estimator is based on performing end-to-end interpolation using gated recurrent unit (GRU) unit. Extensive numerical experiments demonstrate that the developed Bi-GRU estimator significantly outperforms the recently proposed CNN-based estimators in different mobility scenarios, while substantially reducing the overall computational complexity.