chore: use turborepo for type checking (#64455)

api/package.json

@@ -46,6 +46,7 @@
     "@total-typescript/ts-reset": "0.6.1",
     "@types/jsonwebtoken": "9.0.5",
     "@types/lodash-es": "^4.17.12",
+    "@types/node": "^24.10.2",
     "@types/nodemailer": "6.4.21",
     "@types/supertest": "2.0.16",
     "@types/validator": "13.15.10",
@@ -57,6 +58,7 @@
     "prisma": "6.19.1",
     "supertest": "6.3.3",
     "tsx": "4.21.0",
+    "typescript": "5.9.3",
     "vitest": "^3.2.4"
   },
   "engines": {
@@ -82,6 +84,7 @@
     "test": "vitest run",
     "test:watch": "vitest",
     "test:ui": "vitest --ui",
+    "type-check": "tsc --noEmit",
     "prisma": "dotenv -e ../.env prisma",
     "postinstall": "prisma generate",
     "exam-env:generate": "tsx tools/exam-environment/generate/index.ts",