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as in given image, I calculated outputs of shifted inputs by convolution    I wanted to prove, time invariance property of system by convolution integral(i.e equal time shift in input result in equal shift in output) but as calculation shown in figure, I got unexpected result, so what I'm missing?

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If your delayed input is $\hat{x}(t)=x(t-\alpha)$, then $\hat{x}(t-\tau)=x(t-\tau-\alpha)$, and not $x(t-\tau+\alpha)$. From here everything will work out as expected.

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