Turkey's e-Government Gateway (e-Devlet) serves over 68 million registered users with more than 9,200 government services, and is increasingly integrating artificial intelligence into citizen-facing applications such as chatbot assistants and eligibility assessments. However, no structured technical governance infrastructure currently connects high-level AI policy frameworks, such as the EU AI Act, OECD AI Principles, and Turkey's own National AI Strategy, to the operational reality of deploying AI within a centralized e-government platform. We propose GovAI-Pipe, a four-layer governance pipeline designed using Design Science Research methodology that maps the AI model lifecycle to governance checkpoints: (1) pre-deployment validation for bias testing, explainability, and privacy impact assessment; (2) deployment governance for risk-tier classification and approval workflows; (3) runtime monitoring for drift detection, fairness tracking, and human-in-the-loop escalation; and (4) post-incident governance for audit trails, rollback, and citizen redress. Each layer is anchored to specific provisions of the EU AI Act, the GDPR data protection framework, and the National AI Strategy. We demonstrate the framework through two high-risk e-Devlet use cases, showing how GovAI-Pipe operationalizes governance principles as auditable, technical pipeline components.