  We consider the point-to-point continuum directed random polymer
($\mathsf{CDRP}$) model that arises as a scaling limit from $1+1$ dimensional
directed polymers in the intermediate disorder regime. We show that the
annealed law of a point-to-point $\mathsf{CDRP}$ of length $t$ converges to the
Brownian bridge under diffusive scaling when $t \downarrow 0$. In case that $t$
is large, we show that the transversal fluctuations of point-to-point
$\mathsf{CDRP}$ are governed by the $2/3$ exponent. More precisely, as $t$
tends to infinity, we prove tightness of the annealed path measures of
point-to-point $\mathsf{CDRP}$ of length $t$ upon scaling the length by $t$ and
fluctuations of paths by $t^{2/3}$. The $2/3$ exponent is tight such that the
one-point distribution of the rescaled paths converges to the geodesics of the
directed landscape. This point-wise convergence can be enhanced to
process-level modulo a conjecture. Our short and long-time tightness results
also extend to point-to-line $\mathsf{CDRP}$. In the course of proving our main
results, we establish quantitative versions of quenched modulus of continuity
estimates for long-time $\mathsf{CDRP}$ which are of independent interest.
