We propose a new preferential attachment-based network growth model in order to explain two properties of growing networks: (1) the power-law growth of node degrees and (2) the decay of node relevance. In preferential attachment models, the ability of a node to acquire links is affected by its degree, its fitness, as well as its relevance which typically decays over time. After a review of existing models, we argue that they cannot explain the above-mentioned two properties (1) and (2) at the same time. We have found that apart from being empirically observed in many systems, the exponential growth of the network size over time is the key to sustain the power-law growth of node degrees when node relevance decays. We therefore make a clear distinction between the event time and the physical time in our model, and show that under the assumption that the relevance of a node decays with its age $τ$, there exists an analytical solution of the decay function $f_R$ with the form $f_R(τ) = τ^{-1}$. Other properties of real networks such as power-law alike degree distributions can still be preserved, as supported by our experiments. This makes our model useful in explaining and analysing many real systems such as citation networks.