We have fabricated a bridge-type structure for microcalorimeters with superconducting transition-edge sensors (TES). Instead of open space under a SiNx membrane in conventional TES microcalorimeters, the bridge-type calorimeters have a SiTx membrane floating on the Si substrate covered by a SiO2 layer with a small gap of 30-50 mum. The bridge-type structure ensures that the calorimeters are mechanically tough. In addition, the thermal conductance can easily be controlled by changing the width, length, or thickness of the SiNx bridge. The calorimeters, of which operating temperature is 0.43K, consist of a Ti/Au bilayer TES and an Au absorber. The x-ray events were read out by a DC-SQUID current amplifier with a 200-series array of SQUIDs placed on a 4.2 K stage. By analyzing the output pulse shapes, it has been found that the pulses are put into two categories. One has a fast rise time of similar to3 mus and two decay components with time constants of similar to 10 mus and about similar to 130 mus. Another one has a longer rise time of similar to 10 mus and a single decay component of similar to 130 mus. It is considered that the pulse shapes depend on the x-ray absorption positions and the heat-flow pass.