How to compare between two cryptographic algorithms in terms of security? How to compare between two cryptographic algorithms (e.g. SHA-1 and SipHash) in terms of security? That is, how could one prove that algorithm X is more secure than algorithm Y?
TL;DR, you don't.

At this point, we have algorithms we believe are unbroken by current adversaries. For hashing, this includes the SHA-2 family of hashes, SHA-3, BLAKE2b and others. For authentication, we have the HMAC family of functions, the UMAC family of functions, Poly1305, and others. For symmetric encryption, we have AES, ChaCha20, and others. For asymmetric encryption, we have RSA, too many elliptic curves to name, and others. For digital signatures, we have RSA, ECDSA, and others.

When used correctly and with appropriate key sizes, all of these algorithms are considered secure. "More" secure without additional context is a mostly meaningless question: none of these algorithms will ever be brute-forced. Some or all of these algorithms may contain flaws in their design which could render them useless or weakened tomorrow, next year, or a century from now. Some or all of these algorithms may never be broken. But there's no way to know which ones will fall into which category until this happens.

In specific contexts, some algorithms may have desirable properties. The so-called safecurves, for example, are considered more resilient against poorly-written implementations. Block ciphers like AES in virtually all real-world scenarios must be used in a mode of operation that should be chosen based on  the (not necessarily security-based) requirements of your application.

When misused, any algorithm is typically susceptible to catastrophic failure. And unfortunately, virtually any implementation or composition of cryptographic primitives by amateurs (and even, frequently, professionals) is prone to mistakes.

The solution to this is to operate at the highest level of abstraction available to you which is provided by respected cryptographers and software developers. Typically, this means GPG for data at rest, TLS for data in motion. These projects have the benefit of many eyeballs, decades of fixing security issues, and reasonably general default choices for algorithms.

If you're not a seasoned cryptographer and you're making a decision about what cryptographic primitive to use, you have already lost.