WASHINGTON—Republicans advanced legislation through two House committees on Thursday as part of their goal to dismantle the Affordable Care Act, but signs of discord spread around the capital as conservative lawmakers warned this version of the health-law overhaul won’t pass.

On party-line votes, the committees on Energy and Commerce and Ways and Means approved measures repealing major parts of the 2010 health law, known as Obamacare, with the goal of holding a floor vote later this month.

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