OP, I believe that you absolutely can be a relaxed Christian. Having said that, I absolutely understand where you're coming from; I've been in a similar situation myself, and have found it surprising that even where I live(London), there are so few genuinely liberal, inclusive churches that also happen to be thriving. I do think that the Church (not necessarily the Bible or Christianity per se) has a massive issue with women, and with minority groups in general. Often, I think, particular interest groups within the Church use the Bible to their own ends in order to shore up the entrenched hierarchies.
Growing up with no religions, I was first exposed to 'serious' Christianity at university. Even though I found many of the messages appealing at the time, and got quite caught up in the whole thing, I was always uncomfortable with my new friends' attitudes towards gender; I remember a very devout friend of mine agonising over whether she should stand for a post on the committee because she'd be brought up to believe that women should not be in leadership positions. At my university there was even an official ban on women becoming president (this emerged after I left; while I was there it was just accepted that no female student had ever won the ballot). Several male friends ultimately ended up travelling to other countries in search of suitable wives, as they considered British women to be too emancipated!
Back home in London, I attended a local church that a university friend had recommended. It's a large, thriving evangelical church in an affluent middle-class area. Most of the congregations are well-educated professionals, and yet the attitudes I encountered towards women, gay people and divorcees were surprisingly illiberal and backward - women were only allowed to lead groups relating to children (even then, a male youth worker took over once the children became teenagers). When the church built a new hall and meeting centre, the men took on the leadership of the project while their wives worked in the café or ran the crèche. Gay people were 'welcome', providing they remained celibate; as a result I met several long-established members of the congregation who were clearly denying their own sexuality in order to fit in. Sex before marriage was strongly disapproved off, resulting in a steady stream of marriages among young people who were often barely out of their teens. (DP and I, still happily unmarried 10 years on, definitely bucked the trend there!)
The teaching itself was always terribly mild-mannered (and rather convoluted), but there was an insidious overarching message that straight, white, 'happily married' men ran the show. In the end, DP and I left - we're both pretty liberal, and it was all too insular and regressive. There was a mini exodus of the more liberal church members shortly afterwards, and the church was since become deeply conservative in outlook.
Problem is, there seem to be few local alternatives - all the churches nearby are either American-style conservative evangelical (whether middle-class or multi-ethnic 'Mega Churches') or traditional CofE with a dwindling congregation that's shored up by parents who are only attending to get their DC into the local CofE school. I really think there's a gap in the market for a modern church with a liberal, inclusive, non-judgemental ethos, which allows for a healthy level of debate, questioning and reflection and doesn't require its members to leave their brains at the door.