The L3 module is not a pure nn.Sequential module and uses a skip connection, is use_res_connect is defined as seen here.
Adding this skip connection to your code yields the same outputs:
input_data = torch.randn(1, 3, 224, 224)
layer = models.mobilenet_v2(pretrained=True).features[0:4]
layer.eval()
out_original = layer(input_data)

L_zero_from_two = layer[0:3]
output_middle = L_zero_from_two(input_data)

L3 = layer[3]

output = L3.conv[0](output_middle)
output = L3.conv[1](output)
output = L3.conv[2](output)
output = L3.conv[3](output)
if L3.use_res_connect:
    output = output + output_middle

output_test = L3(output_middle)
print((output - output_test).abs().max())
> tensor(0., grad_fn=<MaxBackward1>)
print((out_original - output_test).abs().max())
> tensor(0., grad_fn=<MaxBackward1>)