Secure symmetric encryption algorithm for any-size base62 data
If you define your encryption to be $C=E_{62}(E_k(D_{62}(P)))$ and your decryption to be $P=E_{62}(D_k(D_{62}({C})))$ where $P$ is your encoded URL then you've brought back your problem to finding an encryption function for $l$ bits, where $l$ is the size of $D_{62}(P)$. After that you can "just" look for a Format Preserving Encryption primitive for those $l$ bits. If the encrypted URL is allowed to be larger than the unencrypted URL then you could use CTR mode with a nonce of 8 random bytes.