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Networks are ubiquitous in diverse real-world systems. Many empirical networks grow as the number of nodes increases with time. Percolation transitions in growing random networks can be of infinite order. However, when the growth of large clusters is suppressed under some effects, e.g., the Achlioptas process, the transition type changes to the second order. However, analytical results for the critical behavior, such as the transition point, critical exponents, and scaling relations are rare. Here, we derived them explicitly as a function of a control parameter $m$ representing the suppression strength using the scaling ansatz. We then confirmed the results by solving the rate equation and performing numerical simulations. Our results clearly show that the transition point approaches unity and the order-parameter exponent $β$ approaches zero algebraically as $m \to \infty$, whereas they approach these values exponentially for a static network. Moreover, the upper critical dimension becomes $d_u=4$ for growing networks, whereas it is $d_u=2$ for static ones.