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The exponential distribution (modal value 0) is not bell shaped. If you don't like it's range of non-negative, then take some smooth mollification

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Yes or the chi-square distribution with k=1 or 2 or any other of the gamma distributions[1] with the right parameters will have a shape that is one-sided with the mode at the lower extreme and no "low tail" in the normal sense.

[1] https://en.wikipedia.org/wiki/Gamma_distribution or https://stats.libretexts.org/Bookshelves/Probability_Theory/...

And the smooth mollification will look like...?

in the simplest case... just mirror it (some call this a Laplace distribution). if you don't like how it's not differentiable at the mode there are further smoothings (see, e.g., the wikipedia article for this distribution) but this simple construction is continuous.

It looks like a spike, not a bell.

A spike is not smooth (typically meaning continuous in the variable and its first derivative), which was one of the conditions.

Then take a Cauchy or a t-distribution. Basically anything with a longer tail than exp(x^2). The Gaussian summary will be misleading because of the tails.

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