How does using a Turing-complete programming language compromise the security of the network?
It is much easier to formally verify the behavior space of Bitcoin script vs. Solidity. 

For example, the DAO project suffered from reentrancy vulnerabilities, given that one could not predict the behavior of anonymous smart contracts when they were called/sent funds to (fallback function, now deprecated). 

There are formal verification efforts for Solidity, but this increases the design difficulty tremendously, and one must ask whether it is worth the turing-completeness.