Four reasons off the top of my head.
1. Conservatives don't know how to argue and persuade. In the main, conservatives are not at home on the plane of ideas and abstractions where one must do battle with leftist obfuscation. Conservatives are often non-intellectual when they are not anti-intellectual. I am talking about conservatives 'in the trenches' of ordinary life, politics, and the mass media, the ones with some clout; I am not referring to conservative intellectuals who are intellectual enough but whose influence is limited.
Conservatives, by and large, are doers not thinkers, builders, not scribblers. They are at home on the terra firma of the concrete particular but at sea in the realm of abstraction. The know in their dumb inarticulate way that there is something deeply wrong with same-sex 'marriage,' but they cannot explain what it is in a manner to command the respect of their opponents. George W. Bush, a well-meaning, earnest fellow whose countenance puts me in mind of that of Alfred E. Neuman, could only get the length of such flat-footed asseverations as: "Marriage is between a man and a woman."
That's right, but it is a bare assertion. Sometimes bare assertions are justified, but one must know how to counter those who consider them gratuitous assertions. What is gratuitously asserted may be gratuitously denied without breach of logical propriety, a maxim long enshrined in the Latin tag Quod gratis asseritur, gratis negatur. This is why one reasonably demands arguments from those who make assertions. Arguments are supposed to move us beyond mere assertions and counter-assertions.
Could G. W. Bush present a reasoned defense of traditional marriage, or rather, just plain marriage, against the leftist innovators? If he could he never to my knowledge supplied any evidence that he could.
2. Conservatives muddy the waters with religion, when the topic ought to be approached in a secular way. After all, we need to convince secularists and that will be impossible if we rely on religious traditions and doctrines. You may believe that traditional marriage is an institution with divine sanction. But that will cut no ice with an atheist! Sorry to say something so bloody obvious, but it needs to be said.
Suppose I want to convince you of something. I must use premises that you accept to have any hope of success. For if I mount an argument sporting one or more premises that you do not accept, you will point to that premise or those premises and pronounce my argument unsound no matter how rigorous and cogent my reasoning.
One has to be able to make a secular case for the defense of traditional marriage. The following is the beginning of a secular argument:
One needs to ask about the justification of the state's involvement in marriage in the first place. It is obvious, I hope, that the state ought not be involved in every form of human association. State involvement in any particular type of human association must therefore be justified. We want as much government as we need, but no more. The state is coercive by its very nature, as it must be if it is to be able to enforce its mandates and exercise its legitimate functions, and is therefore at odds with the liberty and autonomy of citizens. It is not obvious that the government should be in the marriage business at all. The burden is on the state to justify its intervention and regulation. But there is a reason for the state to be involved. The state has a legitimate interest in its own perpetuation and maintenance via the production of children, their socializing, their protection, and their transformation into productive citizens who will contribute to the common good. (My use of 'the state' needn't involve an illict hypostatization.) It is this interest that justifies the state's recognition and regulation of marriage as a union of exactly one man and exactly one woman.
I have just specified a reason for state involvement in marriage, one that doesn't rest on any religious premise or assumption. But this justification is absent in the case of same-sex couples since they are not and cannot be productive of children. So here we have a reason why the state ought not recognize same-sex marriage. One and the same biological fact both justifies state regulation and recognition of marriage and justifies the restriction of such recognition to opposite-sex couples. The fact, again, is that only heterosexuals can procreate.
What I have just sketched, if suitably extended, will persuade at least some secularists. Enough to make a difference? I don't know.
3. Liberalism is Emotion-Driven. Just as one cannot hope to persuade people using premises they adamantly reject, one cannot get through to people whose skulls are full of emotional mush. To the emotion-driven, the obviously discriminatory exclusion of same-sex couples from marriage ends the discussion. They won't stay to listen to an explanation as to why some forms of discrimination are justified.
4. The Conservative Disadvantage.