Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving partial differential equations (PDEs) by incorporating physical knowledge into neural network training. When applying the PINN framework to stiff PDEs, a deterioration of training speed and accuracy can be observed. From a mathematical perspective, the high ratio of coefficients in a stiff PDE is the source of locking effects, known from other discretization schemes This work investigates transverse shear locking in PINNs for Timoshenko beams and Reissner-Mindlin plates and proposes effective mitigation strategies. We implement energy-based PINNs using different parametrizations of the kinematics: the standard parametrization and hierarchic parametrizations. Our results demonstrate that standard approaches exhibit severe locking behavior, with training speed deteriorating significantly as the slenderness increases. To address this issue, we develop locking-free PINNs by: (1) using hierarchic formulations that directly parametrize the shear deformation, (2) employing separate neural networks for each primary variable to isolate the trainable parameters, and (3) introducing a slenderness-dependent scaling factor for the shear deformation variable to balance the initial errors in bending and shear energy terms. The proposed method achieves fast, slenderness-independent training and convergence for both beam and plate problems for all variables of interest. The method is implemented for classical, forward-solving PINNs, but can be easily implemented for other methods relying on a PDE-based loss function term such as operator learning approaches.