There is an urgent need to address the transient synchronization stability (TSS) problem caused by the increasing penetration of renewable resources into the power network through grid-following converters (GFL-VSCs). It is challenging to analyze the TSS in power systems with high penetration of renewable resources due to the complex interaction between a large number of GFL-VSCs and power network. Although some research has been carried out on TSS assessment of GFL-VSC systems, its application is limited because measuring the numerous internal parameters of distributed converters is hard. To fill this gap, this paper proposes a TSS analysis method using local network measurements. The present study first constructs a port energy function for PLL-controlled converters, establishing the relationship between the internal control and the transient energy of the converter. Secondly, the branch energy function is constructed using the law of energy conservation. Due to transient energy interactions between the converters and the branches, the transient instability characteristics of the converters are mapped into the network. Transient instability characteristics are extracted through PMUs in the network. On this basis, the TSS of the GFL-VSC systems is analyzed through the identified critical branches. Finally, the effectiveness of the proposed method is validated through simulations on the modified IEEE 39-BUS system and a practical power system.